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MIT_HST512_Genomic_Medicine_Spring_2004 | Lecture_18_Case_Hx_Cancer_Diagnostics.txt | TODD GOLUB: So I come at this actually originally from a pediatric oncology perspective. So I'm going to start by giving examples of a couple of patients that I saw in the Jimmy Fund clinic at the Dana-Farber that were typical. So the first patient was a nine-year-old girl who presented to her pediatrician with-- turn ... |
MIT_HST512_Genomic_Medicine_Spring_2004 | Lecture_17_Direct_Prediction_of_Outcome_Mortality.txt | PETER PARK: For today, I'll just talk a little bit more generally at the beginning about a few observations that I've had. Perhaps a little bit about reliability microarray studies. I'll talk about classification problem in general. And then, I'll talk more about phenotypes. And then review some literature that are wel... |
MIT_HST512_Genomic_Medicine_Spring_2004 | Lecture_15_Microarray_Disease_Classification.txt | STEVEN A. GREENBERG: And did you go through the methods used to classify disease in that or-- here and there, OK. Well, that's what I'm focused on in the next two blocks. And I guess, this is a block of four lectures that are going to focus on this area. And this is the Use of Microarrays for Disease Classification. Oh... |
MIT_HST512_Genomic_Medicine_Spring_2004 | Lecture_13_Case_Hx_Complex_Traits.txt | SCOTT WEISS: So this is an outline of what I'm going to talk about, and we're going to begin by of getting at this question of why complex trait human genetics is so difficult. And then go through each of the steps that you would do if you were actually doing this work. The first question you would get asked on an NIH ... |
MIT_HST512_Genomic_Medicine_Spring_2004 | Lecture_10_Association_with_Markers.txt | MARCO RAMONI: What I'm going to talk about after this little introduction about microarrays is how to analyze this BLAST data. And the principle that I try to present to you is that there is no such a thing as putting your data into a freaking machine and expecting to get an answer. The type of analysis you make is alw... |
MIT_HST512_Genomic_Medicine_Spring_2004 | Lecture_4_Microarray_Massively_Parallel_Measurement.txt | ISAAC SAMUEL KOHANE: Also, I forgot to mention at this point, the output of microarray studies is foreign to basic biology researchers. They're used to looking at three or four or five or 20 numbers and performing some easy analysis in an Excel spreadsheet. But the point-- or doing a BLAST of one gene at a time. But th... |
MIT_HST512_Genomic_Medicine_Spring_2004 | Lecture_9_Machinelearning_Approach.txt | MARCO RAMONI: Today I'm going to talk to you about the-- about basic genetics, what geneticists do, and how genetics is moving into the genomic area by increasing the size, the scope, and the quality of genetic studies. We'll do-- [DOOR CREAKING] You can close the door. So the origin of all of this is-- I will put myse... |
MIT_HST512_Genomic_Medicine_Spring_2004 | Lecture_7_Informational_Resources.txt | ALBERTO RIVA: Alberto Riva, I'm an instructor at CHB. I'm going to talk to you today about the most important resources for finding and using biomedical information, especially information connected with the study of the human genome. So this is going to be something probably slightly different from what you've heard s... |
MIT_HST512_Genomic_Medicine_Spring_2004 | Lecture_2_Introduction_to_Biology_and_Genomic_Measurement.txt | ATUL J. BUTTE: So I can and have in the past talked for about 6 hours on this subject. Today, we're just going to talk about the first of these, microbiology for the [? bioinformaticist. ?] And if we have time, then we can talk about gene measurement techniques, not just microarrays, but all sorts of different technolo... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Inferring_a_Continuous_Random_Variable_from_a_Discrete_Measurement.txt | Hey guys. Welcome back. Today, we're going to be working on a problem that asks you to find the PMF of a function of a random variable. So let's just jump right in. The problem statement gives you the PMF for a random variable called x. So we're told that there's this random variable x that takes on values minus 3, min... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 21_Bayesian_Statistical_Inference_I.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: It involves real ... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Joint_Probability_Mass_Function_PMF_Drill_2.txt | Hey, guys. Welcome back. Today, we're going to do another fun problem, which is a drill problem on joint PMFs. And the goal is that you will feel more comfortable by the end of this problem, manipulating joint PMFs. And we'll also review some ideas about independents in the process. So just to go over what I've drawn h... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | A_Coin_Tossing_Puzzle.txt | Hi. In this problem, we'll be going over practice with the calculation of conditional probabilities. We'll start with a game where our friend Alice will be tossing a coin with certain bias of having a head, and tosses this coin twice. And we're interested in knowing, what's the probability that both coin tosses will en... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Rooks_on_a_Chessboard.txt | Today, we're going to do a fun problem called rooks on a chessboard. And rooks on a chessboard is a problem that's going to test your ability on counting. So hopefully by now in class, you've learned a few tricks to approach counting problems. You've learned about permutations, you've learned about k-permutations, you'... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | A_Chess_Tournament_Problem.txt | Hi. Welcome back. Today, we're going to do a fun problem called the chess tournament problem. Now, it's a very long problem, so I just want to jump straight in. Essentially, the problem statement describes a very special chess tournament, which involves players named Al, Bo, and Chi. Now Al is the current reigning cham... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Competing_Exponentials.txt | Hi, in this problem, we're going to look at competing exponential. So we have three exponential random variables, X with parameter lambda, Y with parameters mu, and Z with parameter nu. And we want to calculate some probability. And the probability that we want to calculate is the probability that X is less than Y is l... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Ambulance_Travel_Time.txt | In this problem, we'll be looking at an ambulance that is traveling back and forth in interval of size l. Say from 0 to l. At some point in time, there's an accident occurring, let's say at location x. And we'll assume the accident occurs in a random location so that x is uniformly distributed between 0 and l. Now, at ... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Conditional_Probability_Example.txt | Hi. Today we're going to do another fun problem that involves rolling two dice. So if you guys happen to frequent casinos, this problem might be really useful for you. I'm just kidding. But in all seriousness, this problem is a good problem, because it's going to remind us how and when to use the discrete uniform law. ... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Using_the_Conditional_Expectation_and_Variance.txt | Hey guys. Welcome back. Today we're going to do a fun problem that will test your knowledge of the law of total variance. And in the process, we'll also get more practice dealing with joint PDFs and computing conditional expectations and conditional variances. So in this problem, we are given a joint PDF for x and y. S... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 18_Markov_Chains_III.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOHN TSITSIKLIS: So what we'... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 7_Discrete_Random_Variables_III.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK, good morning.... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Hypergeometric_Probabilities.txt | In this problem, we're given an urn with n balls in it, out of which m balls are red balls. To visualize it, we can draw a box that represents the set of all n balls. Somewhere in the middle or somewhere else we have a cut, such that to the left we have all the red balls (there are m), and non-red balls. Let's for now ... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Convergence_in_Probability_Example.txt | In this problem, we're given a random variable X which has a uniform distribution in the interval negative 1 to 1. In other words, if we were to draw out the PDF of X, we see that in the interval negative 1 to 1, it has value 1/2. Now we're given a sequence random variables X1, X2, and so on, where each Xi has the sam... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Widgets_and_Crates.txt | Hi. In this problem, we'll get more practice using conditioning to help us calculate expectations of variances. We'll see that in this problem, which deals with widgets and crates, it's actually similar in flavor to an earlier problem that we did, involving breaking a stick twice. And you'll see that in this problem, w... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | A_Random_Walker.txt | In this problem, we'll be working with a object called random walk, where we have a person on the line-- or a tight rope, according to the problem. Let's start from the origin, and each time step, it would randomly either go forward or backward with certain probability. In our case, with probability P, the person would... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Random_Incidence_Under_Erlang_Arrivals.txt | Hi. In this problem, we're going to look at random incidence under Erlang arrivals. First, let's parse what that means. In a Poisson process, remember, the time between arrivals, or the inter-arrival time, is distributed as an exponential random variable. And random incidence for a Poisson process refers to the somewha... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 16_Markov_Chains_I.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So we're going to... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | A_Derived_Distribution_Example.txt | Hi. In this problem we'll work through an example of calculating a distribution for a minute variable using the method of derived distributions. So in general, the process goes as follows. We know the distribution for some random variable X and what we want is the distribution for another random variable of Y, which is... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | A_Mixed_Distribution_Example.txt | In this video, we'll look at an example in which we compute the expectation and cumulative density function of a mixed random variable. The problem is as follows. Al arrives at some bus stand or taxi stand at a given time-- let's say time t equals 0. He finds a taxi waiting for him with probability 2/3 in which he take... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 9_Multiple_Continuous_Random_Variables.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOHN TSITSIKLIS: OK let's st... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | The_Probability_Distribution_Function_PDF_of_X.txt | Hi, In this problem, we'll be looking at the PDF the absolute value of x. So if we know a random variable, x, and we know it's PDF, how can we use that information to help us find the PDF of another random variable-- the absolute value of x? And so throughout this problem, we'll define a new random variable called y. A... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | An_Inference_Example.txt | Hi. In this session, we're going to cover a nice review problem that will look at how to infer one random variable based on another. And in this problem, we're given two random variables-- X and Y-- and we're also given their joint pdf, which we're told is a constant 2/3 within the region bounded by these orange lines.... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 12_Iterated_Expectations.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOHN TSITSIKLIS: So today we... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 17_Markov_Chains_II.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality, educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right. So to... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | The_Difference_of_Two_Independent_Exponential_Random_Variables.txt | In this problem, Romeo and Juliet are to meet up for a date, where Romeo arrives at time x and Juliet at time y, where x and y are independent exponential random variables, with parameters lambda. And we're interested in knowing the difference between the two times of arrivals, we'll call it z, written as x minus y. An... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 10_Continuous_Bayes_Rule_Derived_Distributions.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So today's agenda... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Setting_Up_a_Markov_Chain.txt | Hi. In this problem, we're going to practice setting up a Markov chain by going fishing in this lake, which has n fish in it, some of which are green. And the rest of the fish are blue. So, what we do is, every day we go to this lake, and we catch exactly 1 fish. And all the fish are equally likely to be the 1 that's c... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Calculating_a_Cumulative_Distribution_Function_CDF.txt | Hi. In this problem, we'll get some practice working with PDFs and also using PDFs to calculate CDFs. So the PDF that we're given in this problem is here. So we have a random variable, z, which is a continuous random variable. And we're told that the PDF of this random variable, z, is given by gamma times 1 plus z squa... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Uniform_Probabilities_on_a_Square.txt | In this problem, we will be helping Romeo and Juliet meet up for a date. And in the process, also we'll review some concepts in basic probability theory, including sample spaces and probability laws. This problem, the basic setup is that Romeo and Juliet are trying to meet up for a date. And let's say they're trying to... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Inferring_a_Parameter_of_Uniform_Part_1.txt | Hi. In this problem, Romeo and Juliet are back and they're still looking to meet up for a date. Remember, the last time we met up with them, it was back in the beginning of the course and they were trying to meet up for a date but they weren't always punctual. So we modeled their delay as uniformly distributed between ... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 6_Discrete_Random_Variables_II.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK so let's start... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 19_Weak_Law_of_Large_Numbers.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOHN TSITSIKLIS: We're going... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Convergence_in_Probability_and_in_the_Mean_Part_1.txt | In this exercise, we'll be working with the notion of convergence in probability, as well as some other notion of converge of random variables that we'll introduce later. First type of random variable is xn, where xn has probability 1 minus 1 minus over n to be as 0 and probability of 1 over n to be a 1. And graphical... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | The_Variance_in_the_Stick_Breaking_Problem.txt | Hi. In this problem, we'll get a chance to see the usefulness of conditioning in helping us to calculate quantities that would otherwise be difficult to calculate. Specifically, we'll be using the law of iterated expectations and the law of total variance. Before we get started, let's just take a quick moment to interp... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 20_Central_Limit_Theorem.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: We're going to fi... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | A_Random_Number_of_Coin_Flips.txt | Hey, everyone. Welcome back. Today, we're going to do another fun problem that has to do with a random number of coin flips. So the experiment we're going to run is as follows. We're given a fair six-sided die, and we roll it. And then we take a fair coin, and we flip it the number of times indicated by the die. That i... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Convergence_in_Probability_and_in_the_Mean_Part_2.txt | For part E and F of the problem, we'll be introducing a new notion of convergence, so-called the convergence E mean squared sense. We say that xn converges to a number c in mean squared, if as we take and go to infinity, the expected value of xn minus c squared goes to 0. To get a sense of what this looks like, let's s... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | The_Sum_of_Discrete_and_Continuous_Random_Variables.txt | In this video, we're going to do an example in which we derive the probability density function of the sum of two random variables. The problem tells us the following. We're given that X and Y are independent random variables. X is a discrete random variable with PMF Px. Y is continuous with PDF Fy. And we'd like to co... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 13_Bernoulli_Process.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So by now you hav... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Normal_Probability_Calculation.txt | Hi. In this video, we're going to do standard probability calculations for normal random variables. We're given that x is standard normal with mean 0 and variance 1. And y is normal with mean one and variance 4. And we're asked for a couple of probabilities. For the normal CDF, we don't have a closed form expression. A... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Inferring_a_Parameter_of_Uniform_Part_2.txt | Welcome back. So now we're going to finish the rest of this problem. For part e, we've calculated what the map and LMS estimators are. And now we're going to calculate what the conditional mean squared error is. So it's a way to measure how good these estimators are. So let's start out generically. For any estimator th... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Uniform_Probabilities_on_a_Triangle.txt | Hi. In this problem, we're going to get a bunch of practice working with multiple random variables together. And so we'll look at joint PDFs, marginal PDFs, conditional PDFs, and also get some practice calculating expectations as well. So the problem gives us a pair of random variables-- x and y. And we're told that th... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Geniuses_and_Chocolates.txt | Hi. Today, we're going to do a really fun problem called geniuses and chocolates. And what this problem is exercising is your knowledge of properties of probability laws. So let me just clarify what I mean by that. Hopefully, by this point, you have already learned what the axioms of probability are. And properties of ... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | The_Coupon_Collector_Problem.txt | In this exercise, we'll be looking at a problem, also know as the coupons collector's problem. We have a set of K coupons, or grades in our case. And each time slot we're revealed with one random grade. And we'd like to know how long it would take for us to collect all K grades. In our case, K is equal to 6. Now the ke... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Mean_First_Passage_and_Recurrence_Times.txt | In this problem, we are looking at a student whose performance from day to day sort of oscillates according to a Markov chain. In particular, the student can either be in state 1, which is a state of being up to date, or in state 2, which is a state of being kind of fallen behind. Now, the transition probabilities betw... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 25_Classical_Inference_III.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu PROFESSOR: OK, if you have no... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Probability_that_Three_Pieces_Form_a_Triangle.txt | In this problem, we're going to look at the probability that when you take a stick and break it into three pieces randomly that these three pieces can actually be used to form a triangle. All right, so we start out with a stick of unit length, so-- length 1. And we'll choose a point along the stick to break. And we'll ... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | The_Monty_Hall_Problem.txt | Hi. In the session, we'll be solving the Monty Hall problem. And this problem is based on an old game show that was called "Let's Make a Deal." And the host of this game show, his name was Monty Hall, which is why this problem is now known as the Monty Hall problem. And this problem is actually pretty well-known, becau... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 8_Continuous_Random_Variables.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOHN TSITSIKLIS: OK. We can ... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Joint_Probability_Mass_Function_PMF_Drill_1.txt | Welcome back guys. Today we're going to work on a problem that tests your knowledge of joint PMFs. And we're also going to get some practice computing conditional expectations and conditional variances. So in this problem, we are given a set of points in the xy plane. And we're told that these points are equally likely... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Sampling_People_on_Buses.txt | Hi. In this problem, we're dealing with buses of students going to a job convention. And in the problem, we'll be exercising our knowledge of PMFs-- probability mass functions. So we'll get a couple of opportunities to write out some PMFs, and also calculating expectations or expected values. And also, importantly, we'... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Bernoulli_Process_Practice.txt | Hi everyone. Today I'm going to talk about Bernoulli process practice number one. In this problem, you are visiting a rain forest. But unfortunately you have run out of insect repellent. As a result, the probability of you getting mosquito bites is really high. At each second, the probability that a mosquito will land ... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 3_Independence.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Let us start. So ... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 23_Classical_Statistical_Inference_I.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality, educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So for the last ... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 15_Poisson_Process_II.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOHN TSITSIKLIS: Today we're... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 22_Bayesian_Statistical_Inference_II.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu PROFESSOR: So we're going to ... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Inferring_a_Discrete_Random_Variable_from_a_Continuous_Measurement.txt | Hi. In this problem, we're going to look at how to infer a discrete random variable from a continuous measurement. And really, what it's going to give us is some practice working with a variation of Bayes' rule. So the problem tells us that we have a discrete random variable x with this PMF. It is 1 with probability P,... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Using_the_Central_Limit_Theorem.txt | Hi. In this video, we're going to do some approximate calculations using the central limit theorem. We're given that Xn is the number of gadgets produced on day n by a factory. And it has a normal distribution with mean 5 and variance 9. And they're all independent and identically distributed. We're looking for the pro... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Markov_Chain_Practice_1.txt | Hi, everyone. Today, I'm going to talk about Markov Chain Practice number one. Before we start, let's first take a look at this Markov chain. This Markov chain has six states. In this problem, we always assume the process starts from state S0. On the first trial, the process can either make a transition from S0 to S1 w... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Probabilty_Bounds.txt | In this problem, we're given a collection of 10 variables, x1 through x10, where each i, xi, is a uniform random variable between 0 and 1. So each i is uniform between 0 and 1, and all 10 variables are independent. And we'd like to develop a bound on the probability that some of the 10 variables, 1 to 10, being greater... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | A_Coin_with_Random_Bias.txt | Hi. In this problem, we're going to be dealing with a variation of the usual coin-flipping problem. But in this case, the bias itself of the coin is going to be random. So you could think of it as, you don't even know what the probability of heads for the coin is. So as usual, we're still taking one coin and we're flip... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 4_Counting.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK. So today's l... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Communication_over_a_Noisy_Channel.txt | Hi. In this problem, we'll be talking about communication across a noisy channel. But before we dive into the problem itself, I wanted to first motivate the context a little bit and talk more about what exactly a communication channel is and what "noise" means. So in our everyday life, we deal with a lot of communicati... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 24_Classical_Inference_II.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOHN TSITSIKLIS: And we're g... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 14_Poisson_Process_I.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So last time we s... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | The_Absent_Minded_Professor.txt | Hi. In this problem, we have an absent-minded professor who will inadvertently give us some practice with exponential random variables. So the professor has made two appointments with two students and inadvertently made them at the same time. And what we do is we model the duration of these appointments with an exponen... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 2_Conditioning_and_Bayes_Rule.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu JOHN TSISIKLIS: So here's the... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 5_Discrete_Random_Variables_I.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu, OK. So let us start. All ri... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 1_Probability_Models_and_Axioms.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK, so welcome to... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | The_Probability_of_the_Difference_of_Two_Events.txt | Hi. In this problem, we're going to use the set of probability axioms to derive the probability of the difference of two events. Now, before we get started, there's one thing you might notice that, the equation we're trying to prove is actually quite complicated. And I don't like it either, so the first thing we're goi... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Flipping_a_Coin_a_Random_Number_of_Times.txt | In this problem, we're looking at a two stage process in which the first stage, we roll a fair die which has four faces to obtain a number N, where N belongs to the set 0, 1, 2, and 3 with equal probability. Now, given the result of the die roll, N will toss a fair coin N times in getting K heads from the coin tosses. ... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | PMF_of_a_Function_of_a_Random_Variable.txt | Hey guys. Welcome back. Today, we're going to be working on a problem that asks you to find the PMF of a function of a random variable. So let's just jump right in. The problem statement gives you the PMF for a random variable called x. So we're told that there's this random variable x that takes on values minus 3, min... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | 11_Derived_Distributions_ctd_Covariance.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality, educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Good morning. So... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Network_Reliability.txt | Previously, we learned the concept of independent experiments. In this exercise, we'll see how the seemingly simple idea of independence can help us understand the behavior of quite complex systems. In particular, we'll combined the concept of independence with the idea of divide and conquer, where we break a larger sy... |
MIT_6041SC_Probabilistic_Systems_Analysis_and_Applied_Probability_Fall_2013 | Mean_Variance_of_the_Exponential.txt | Hi. In this video, we're going to compute some useful quantities for the exponential random variable. So we're given that x is exponential with rate lambda. PDF looks like this, and the formula is here. First question, part a, what's the CDF? So let's go right in. The CDF of x is the probability that X is less than or ... |
SLAM_Lectures | SLAM_A_03.txt | so now we had a look at the motor Tis of the robot but what we want to know is where the robot is so we need to know how the robot ticks translate into a movement of the robot so we need a motion model let's have a look at the robot once again this is the robot has seen from above and these are the caterpillar tracks o... |
SLAM_Lectures | SLAM_B_05.txt | and now the next step is to use the correspondences in order to estimate the transformation and this is so trivial that I implemented that already for you it's called slam 5p estimated wall transform as you see down here in main here's the loop over all positions and this is exactly the same as before and here's the di... |
SLAM_Lectures | PP_08.txt | so we will now do a final modification to our AAR algorithm and this is due to the following observation say you want to go from this start to this goal and you're driving on a parking lot for example and so here's another parked car and here's yet another car then as you see and as we expect from our algorithm those c... |
SLAM_Lectures | SLAM_C_03.txt | so let's check it out so it ends at 200 or 2001 or 199 with a probability of 0.25 0.5 and 0.25 and the previous 99 result may now in the Second Step move exactly for 1 meter with a probability of 0.5 might also overshoot with probability of 0.25 or undershoot probability of 0.25 and end up in 198 cm and the same holds ... |
SLAM_Lectures | SLAM_E_03.txt | now here's the second question remember the calman filter in our last unit so we had this Arena we had our landmarks and we had our uncertainty model by a covariance matrix and then we predicted and corrected and predicted and corrected and so on so now say initially we don't know our Precision so we just say it's a ve... |
SLAM_Lectures | SLAM_B_01.txt | welcome to Unit B of our slam lecture and this will be about using sensor data or measurements to improve the robot state so if you remember the last time we did two things first of all we computed the robot's trajectory so we used the motor signals to determine when the robot goes straight and how fast it goes straigh... |
SLAM_Lectures | PP_07.txt | and this can be seen as follows if you started here I want you go to here then the algorithm will proceed walking towards the guild but now say there is an obstacle which looks like that then the algorithm will still try to walk towards to go and indeed the distance to the goal is always decreasing now if this obstacle... |
SLAM_Lectures | SLAM_B_04.txt | let's review what we did so far so starting from scan with many many points we took the approach to identify a certain subset of points namely those close to large jumps in the range values and from that computed positions of cylinders assuming that those combinations of a falling and rising edge belong to cylinders in... |
SLAM_Lectures | SLAM_F_01.txt | now welcome to unit F and this will be about simultaneous localization and mapping which is the topic that gave the slam lecture its name now let's have a look at what we did so far so here's our robot at a given position having a certain orientation and we also know there is an error associated with position and orien... |
SLAM_Lectures | SLAM_D_16.txt | so when you run your python script it will generate this common prediction. text now open this in the lock file viewer and you will see our well-known trajectory which is no surprise because we now implemented just the prediction step which is not different from what we did before however in addition to Computing the s... |
SLAM_Lectures | PP_10.txt | to see what happens let us compare our previous version of AAR to our new implementation using the kinematic State space of the vehicle so in our previous implementation we had a roster of cells so when we were in one cell we could explore all the other cells and check if one of those neighbors has to be put into our f... |
SLAM_Lectures | SLAM_D_13.txt | now let's apply all this to our robot well first of all we need a motion model but as you remember in Unit A we already set up the motion model for our robot there's a track withd W and there's some left and right movement off the tracks and by that the robot will move on a curve segment where the radius are and after ... |
SLAM_Lectures | SLAM_G_06.txt | so now finally as a last modification to our fast slam algorithm we're looking for a way to get rid of those landmarks that appear at some point in time but then are not measured subsequently and still stay in the list of landmarks in our particles now here is an approach to deal with those spurious landmarks so when o... |
SLAM_Lectures | SLAM_G_04.txt | now let's have a look at the third part the landmark update so now the situation is as follows a robot makes a measurement identifies a cylinder at a certain range and bearing angle and there's an existing cylinder some covariance matrix and after computing the likelihoods we decide that this measurement belongs to tha... |
SLAM_Lectures | SLAM_B_06.txt | and this is the outcome of our ICP algorithm it generates the ICP of all transform the text so open this and here's the result as you see the trajectory of our robot is much smoother than the previous result and as we step through the robots positions we see now that by using the ICP those scan points that were current... |
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