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MIT_RES6012_Introduction_to_Probability_Spring_2018 | L033_Independence_of_Two_Events.txt | In the previous example, we had a model where the result of the first coin toss did not affect the probabilities of what might happen in the second toss. This is a phenomenon that we call independence and which we now proceed to define. Let us start with a first attempt at the definition. We have an event, B, that has ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L016_More_Properties_of_Probabilities.txt | We will now continue and derive some additional properties of probability laws which are, again, consequences of the axioms that we have introduced. The first property is the following. If we have two sets and one set is smaller than the other-- so we have a picture as follows. We have our sample space. And we have a c... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L062_Variance.txt | We have introduced the concept of expected value or mean, which tells us the average value of a random variable. We will now introduce another quantity, the variance, which quantifies the spread of the distribution of a random variable. So consider a random variable with a given PMF, for example like the PMF shown in t... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L044_Combinations.txt | Let us now study a very important counting problem, the problem of counting combinations. What is a combination? We start with a set of n elements. And we're also given a non-negative integer, k. And we want to construct or to choose a subset of the original set that has exactly k elements. In different language, we wa... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S013_Sequences_and_their_Limits.txt | In this segment, we will discuss what a sequence is and what it means for a sequence to converge. So a sequence is nothing but some collection of elements that are coming out of some set, and that collection of elements is indexed by the natural numbers. We often use the notation, and we say that we have a sequence ai,... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L063_The_Variance_of_the_Bernoulli_The_Uniform.txt | In this segment, we will go through the calculation of the variances of some familiar random variables, starting with the simplest one that we know, which is the Bernoulli random variable. So let X take values 0 or 1, and it takes a value of 1 with probability p. We have already calculated the expected value of X, and ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S018_Countable_and_Uncountable_Sets.txt | Probability models often involve infinite sample spaces, that is, infinite sets. But not all sets are of the same kind. Some sets are discrete and we call them countable, and some are continuous and we call them uncountable. But what exactly is the difference between these two types of sets? How can we define it precis... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L014_Probability_Axioms.txt | We have so far discussed the first step involved in the construction of a probabilistic model. Namely, the construction of a sample space, which is a description of the possible outcomes of a probabilistic experiment. We now come to the second and much more interesting part. We need to specify which outcomes are more l... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S091_Buffons_Needle_Monte_Carlo_Simulation.txt | PROFESSOR: In this segment, we will look at the famous example, which was posed by Comte de Buffon-- a French naturalist-- back in the 18th century. And it marks the beginning of a subject that is known as the subject of geometric probability. The problem is pretty simple. We have the infinite plane, and we draw lines ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S0110_Bonferronis_Inequality.txt | PROFESSOR: In this segment, we will discuss a little bit the union bound and then discuss a counterpart, which is known as the Bonferroni inequality. Let us start with a story. Suppose that we have a number of students in some class. And we have a set of students that are smart, let's call that set a1. So this is the s... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L0510_The_Expected_Value_Rule.txt | In this segment, we discuss the expected value rule for calculating the expected value of a function of a random variable. It corresponds to a nice formula that we will see shortly, but it also involves a much more general idea that we will encounter many times in this course, in somewhat different forms. Here's what i... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L094_Memorylessness_of_the_Exponential_PDF.txt | We now revisit the exponential random variable that we introduced earlier and develop some intuition about what it represents. We do this by establishing a memorylessness property, similar to the one that we established earlier in the discrete case for the geometric PMF. Suppose that it is known that light bulbs have a... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L097_Joint_PDFs.txt | In this segment, we start a discussion of multiple continuous random variables. Here are some objects that we're already familiar with. But exactly as in the discrete case, if we are dealing with two random variables, it is not enough to know their individual PDFs. We also need to model the relation between the two ran... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L096_Mixed_Random_Variables.txt | We now look at an example similar to the previous one, in which we have again two scenarios, but in which we have both discrete and continuous random variables involved. You have $1 and the opportunity to play in the lottery. With probability 1/2, you do nothing and you're left with the dollar that you started with. Wi... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L032_A_Coin_Tossing_Example.txt | As an introduction to the main topic of this lecture sequence, let us go through a simple example and on the way, review what we have learned so far. The example that we're going to consider involves three tosses of a biased coin. It's a coin that results in heads with probability p. We're going to make this a little m... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L045_Binomial_Probabilities.txt | The coefficients n-choose-k that we calculated in the previous segment are known as the binomial coefficients. They are intimately related to certain probabilities associated with coin tossing models, the so-called binomial probabilities. This is going to be our subject. We consider a coin which we toss n times in a ro... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L087_Cumulative_Distribution_Functions.txt | We have seen that several properties, such as, for example, linearity of expectations, are common for discrete and continuous random variables. For this reason, it would be nice to have a way of talking about the distribution of all kinds of random variables without having to keep making a distinction between the diffe... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L083_Uniform_Piecewise_Constant_PDFs.txt | Let us now give an example of a continuous random variable-- the uniform random variable. It is patterned after the discrete random variable. Similar to the discrete case, we will have a range of possible values. In the discrete case, these values would be the integers between a and b. In the continuous case, any real ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S051_Supplement_Functions.txt | We are defining a random variable as a real valued function on the sample space. So this is a good occasion to make sure that we understand what a function is. To define a function, we start with two sets. One set-- call it A-- is the domain of the function. And we have our second set. Then a function is a rule that fo... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L011_Lecture_Overview.txt | Welcome to the first lecture of this class. You may be used to having a first lecture devoted to general comments and motivating examples. This one will be different. We will dive into the heart of the subject right away. In fact, today we will accomplish a lot. By the end of this lecture, you will know about all of th... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L022_Conditional_Probabilities.txt | Conditional probabilities are probabilities associated with a revised model that takes into account some additional information about the outcome of a probabilistic experiment. The question is how to carry out this revision of our model. We will give a mathematical definition of conditional probabilities, but first let... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L065_Total_Expectation_Theorem.txt | An important reason why conditional probabilities are very useful is that they allow us to divide and conquer. They allow us to split complicated probability modes into simpler submodels that we can then analyze one at a time. Let me remind you of the Total Probability Theorem that has his particular flavor. We divide ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L089_Calculation_of_Normal_Probabilities.txt | We have claimed that normal random variables are very important, and therefore we would like to be able to calculate probabilities associated with them. For example, given a normal random variable, what is the probability that it takes a value less than 5? Unfortunately, there are no closed form expressions that can he... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L061_Lecture_Overview.txt | In the previous lecture we introduced random variables, probability mass functions and expectations. In this lecture we continue with the development of various concepts associated with random variables. There will be three main parts. In the first part we define the variance of a random variable, and calculate it for ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L075_Example.txt | Let us now consider a simple example. Let random variables X and Y be described by a joint PMF which is the one shown in this table. Question-- are X and Y independent? We can try to answer this question by using the definition of independence. But it is actually more instructive to proceed in a somewhat more intuitive... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S072_The_Variance_of_the_Geometric.txt | In this segment, we will derive the formula for the variance of the geometric PMF. The argument will be very much similar to the argument that we used to drive the expected value of the geometric PMF. And it relies on the memorylessness properties of geometric random variables. So let X be a geometric random variable w... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L091_Lecture_Overview.txt | In this lecture, we continue our discussion of continuous random variables. We will start by bringing conditioning into the picture and discussing how the PDF of a continuous random variable changes when we are told that a certain event has occurred. We will take the occasion to develop counterparts of some of the tool... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L0511_Linearity_of_Expectations.txt | We end this lecture sequence with the most important property of expectations, namely linearity. The idea is pretty simple. Suppose that our random variable, X, is the salary of a random person out of some population. So that we can think of the expected value of X as the average salary within that population. And now ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L024_Conditional_Probabilities_Obey_the_Same_Axioms.txt | I now want to emphasize an important point. Conditional probabilities are just the same as ordinary probabilities applied to a different situation. They do not taste or smell or behave any differently than ordinary probabilities. What do I mean by that? I mean that they satisfy the usual probability axioms. For example... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L038_Independence_Versus_Pairwise_Independence.txt | We will now consider an example that illustrates the difference between the notion of independence of a collection of events and the notion of pairwise independence within that collection. The model is simple. We have a fair coin which we flip twice. So at each flip, there is probability 1/2 of obtaining heads. Further... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L068_Linearity_of_Expectations_The_Mean_of_the_Binomial.txt | Let us now revisit the subject of expectations and develop an important linearity property for the case where we're dealing with multiple random variables. We already have a linearity property. If we have a linear function of a single random variable, then expectations behave in a linear fashion. But now, if we have mu... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S016_The_Geometric_Series.txt | One particular series that shows up in many applications, examples, or problems is the geometric series. [In] the geometric series, we are given a certain number, alpha, and we want to sum all the powers of alpha, starting from the 0th power, which is equal to 1, the first power, and so on, and this gives us an infinit... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L058_Expectation.txt | Our discussion of random variable so far has involved nothing but standard probability calculations. Other than using the PMF notation, we have done nothing new. It is now time to introduce a truly new concept that plays a central role in probability theory. This is the concept of the expected value or expectation or m... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L098_From_The_Joint_to_the_Marginal.txt | In the discrete case, we saw that we could recover the PMF of X and the PMF of Y from the joint PMF. Indeed, the joint PMF is supposed to contain a complete probabilistic description of the two random variables. It is their probability law, and any quantity of interest can be computed if we know the joint. Things are s... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L092_Conditioning_A_Continuous_Random_Variable_on_an_Event.txt | In this segment, we pursue two themes. Every concept has a conditional counterpart. We know about PDFs, but if we live in a conditional universe, then we deal with conditional probabilities. And we need to use conditional PDFs. The second theme is that discrete formulas have continuous counterparts in which summations ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L0110_Interpretations_Uses_of_Probabilities.txt | We end this lecture sequence by stepping back to discuss what probability theory really is and what exactly is the meaning of the word probability. In the most narrow view, probability theory is just a branch of mathematics. We start with some axioms. We consider models that satisfy these axioms, and we establish some ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L013_Sample_Space_Examples.txt | Let us now look at some examples of sample spaces. Sample spaces are sets. And a set can be discrete, finite, infinite, continuous, and so on. Let us start with a simpler case in which we have a sample space that is discrete and finite. The particular experiment we will be looking at is the following. We take a very sp... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S071_The_InclusionExclusion_Formula.txt | In this segment, we develop the inclusion-exclusion formula, which is a beautiful generalization of a formula that we have seen before. Let us look at this formula and remind ourselves what it says. If we have two sets, A1 and A2, and we're interested in the probability of their union, how can we find it? We take the p... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L099_Continuous_Analogs_of_Various_Properties.txt | In this segment we will go very fast through a few definitions and facts that remain true in the continuous case. Everything is completely analogous to the discrete case. And there are absolutely no surprises here. So, for example, we have defined joint PMFs for the case of more than two discrete random variables. And ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L056_Binomial_Random_Variables.txt | The next random variable that we will discuss is the binomial random variable. It is one that is already familiar to us in most respects. It is associated with the experiment of taking a coin and tossing it n times independently. And at each toss, there is a probability, p, of obtaining heads. So the experiment is comp... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L077_Independence_Variances_the_Binomial_Variance.txt | Let us now revisit the variance and see what happens in the case of independence. Variances have some general properties that we have already seen. However, since we often add random variables, we would like to be able to say something about the variance of the sum of two random variables. Unfortunately, the situation ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L012_Sample_Space.txt | Putting together a probabilistic model-- that is, a model of a random phenomenon or a random experiment-- involves two steps. First step, we describe the possible outcomes of the phenomenon or experiment of interest. Second step, we describe our beliefs about the likelihood of the different possible outcomes by specify... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L0310_The_Kings_Sibling.txt | Let us now conclude with a fun problem, which is also a little bit of a puzzle. We are told that the king comes from a family of two children. What is the probability that his sibling is female? Well, the problem is too loosely stated, so we need to start by making some assumptions. First, let's assume that we're deali... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L048_Each_Person_Gets_An_Ace.txt | We will now apply our multinomial formula for counting the number of partitions to solve the following probability problem. We have a standard 52-card deck, which we deal to four persons. Each person gets 13 cards as, for example, in bridge. What is the probability that each person gets exactly one ace? Well, before we... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S012_De_Morgans_Laws.txt | We will now discuss De Morgan's laws that are some very useful relations between sets and their complements. One of the De Morgan's laws takes this form. If we take the intersection of two sets and then take the complement of this intersection, what we obtain is the union of the complements of the two sets. Pictorially... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L073_Conditional_Expectation_the_Total_Expectation_Theorem.txt | We will now talk about conditional expectations of one random variable given another. As we will see, there will be nothing new here, except for older results but given in new notation. Any PMF has an associated expectation. And so conditional PMFs also have associated expectations, which we call conditional expectatio... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S010_Mathematical_Background_Overview.txt | In this sequence of segments, we review some mathematical background that will be useful at various places in this course. Most of what is covered, with the exception of the last segment, is material that you may have seen before. But this could still be an opportunity to refresh some of these concepts. I should say th... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L026_The_Multiplication_Rule.txt | As promised, we will now start developing generalizations of the different calculations that we carried out in the context of the radar example. The first kind of calculation that we carried out goes under the name of the multiplication rule. And it goes as follows. Our starting point is the definition of conditional p... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L027_Total_Probability_Theorem.txt | Let us now revisit the second calculation that we carried out in the context of our earlier example. In that example, we calculated the total probability of an event that can occur under different scenarios. And it involves the powerful idea of divide and conquer where we break up complex situations into simpler pieces... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L021_Lecture_Overview.txt | Suppose I look at the registry of residents of my town and pick a person at random. What is the probability that this person is under 18 years of age? The answer is about 25%. Suppose now that I tell you that this person is married. Will you give the same answer? Of course not. The probability of being less than 18 yea... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S019_Proof_That_a_Set_of_Real_Numbers_is_Uncountable.txt | For those of you who are curious, we will go through an argument that establishes that the set of real numbers is an uncountable set. It's a famous argument known as Cantor's diagonalization argument. Actually, instead of looking at the set of all real numbers, we will first look at the set of all numbers, x, that belo... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L041_Lecture_Overview.txt | A basketball coach has 20 players available. Out of them, he needs to choose five for the starting lineup, and seven who would be sitting on the bench. In how many ways can the coach choose these 5 plus 7 players? It is certainly a huge number, but what exactly is it? In this lecture, we will learn how to answer questi... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L059_Elementary_Properties_of_Expectation.txt | We now note some elementary properties of expectations. These will be some properties that are extremely natural and intuitive, but even so, they are worth recording. The first property is the following. If you have a random variable which is non-negative, then its expected value is also non-negative. What does it mean... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S073_Independence_of_Random_Variables_Versus_Independence_of_Events.txt | By now, we have defined the notion of independence of events and also the notion of independence of random variables. The two definitions look fairly similar, but the details are not exactly the same, because the two definitions refer to different situations. For two events, we know what it means for them to be indepen... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L036_Independence_Versus_Conditional_Independence.txt | We have already seen an example in which we have two events that are independent but become dependent in a conditional model. So that [independence] and conditional independence is not the same. We will now see another example in which a similar situation is obtained. The example is as follows. We have two possible coi... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L017_A_Discrete_Example.txt | Let us now move from the abstract to the concrete. Recall the example that we discussed earlier where we have two rolls of a tetrahedral die. So there are 16 possible outcomes illustrated in this diagram. To continue, now we need to specify a probability law, some kind of probability assignment. To keep things simple, ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L076_Independence_Expectations.txt | When we have independence, does anything interesting happen to expectations? We know that, in general, the expected value of a function of random variables is not the same as applying the function to the expected values. And we also know that there are some exceptions where we do get equality. This is the case where we... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L078_The_Hat_Problem.txt | We will now study a problem which is quite difficult to approach in a direct brute force manner but becomes tractable once we break it down into simpler pieces using several of the tricks that we have learned so far. And this problem will also be a good opportunity for reviewing some of the tricks and techniques that w... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L042_The_Counting_Principle.txt | In this segment we introduce a simple but powerful tool, the basic counting principle, which we will be using over and over to deal with counting problems. Let me describe the idea through a simple example. You wake up in the morning and you find that you have in your closet 4 shirts, 3 ties, and 2 jackets. In how many... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L074_Independence_of_Random_Variables.txt | We now come to a very important concept, the concept of independence of random variables. We are already familiar with the notion of independence of two events. We have the mathematical definition, and the interpretation is that conditional probabilities are the same as unconditional ones. Intuitively, when you are tol... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S017_About_the_Order_of_Summation_in_Series_with_Multiple_Indices.txt | We now continue our discussion of infinite series. Sometimes we have to deal with series where the terms being added are indexed by multiple indices, as in this example here. We're given numbers, aij, and i ranges over all the positive integers. j also ranges over all the positive integers. So what does this sum repres... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S011_Sets.txt | In this segment, we will talk about sets. I'm pretty sure that most of what I will say is material that you have seen before. Nevertheless, it is useful to do a review of some of the concepts, the definitions, and also of the notation that we will be using. So what is a set? A set is just a collection of distinct eleme... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L018_A_Continuous_Example.txt | We will now go through a probability calculation for the case where we have a continuous sample space. We revisit our earlier example in which we were throwing a dart into a square target, the square target being the unit square. And we were guaranteed that our dart would fall somewhere inside this set. So our sample s... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L025_A_Radar_Example_and_Three_Basic_Tools.txt | Let us now examine what conditional probabilities are good for. We have already discussed that they are used to revise a model when we get new information, but there is another way in which they arise. We can use conditional probabilities to build a multi-stage model of a probabilistic experiment. We will illustrate th... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L085_Mean_Variance_of_the_Uniform.txt | As an example of a mean-variance calculation, we will now consider the continuous uniform random variable which we have introduced a little earlier. This is the continuous analog of the discrete uniform, for which we have already seen formulas for the corresponding mean and variance. So let us now calculate the mean or... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L031_Lecture_Overview.txt | In this lecture, we introduce and develop the concept of independence between events. The general idea is the following. If I tell you that a certain event A has occurred, this will generally change the probability of some other event B. Probabilities will have to be replaced by conditional probabilities. But if the co... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L081_Lecture_Overview.txt | In this lecture, we start our discussion of continuous random variables. We will focus on the case of a single continuous random variable, and we'll describe its distribution using a so-called probability density function, an object that will replace the PMFs from the discrete case. We will then proceed to define the e... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L015_Simple_Properties_of_Probabilities.txt | The probability axioms are the basic rules of probability theory. And they are surprisingly few. But they imply many interesting properties that we will now explore. First we will see that what you might think of as missing axioms are actually implied by the axioms already in place. For example, we have an axiom that p... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L019_Countable_Additivity.txt | We have seen so far an example of a probability law on a discrete and finite sample space as well as an example with an infinite and continuous sample space. Let us now look at an example involving a discrete but infinite sample space. We carry out an experiment whose outcome is an arbitrary positive integer. As an exa... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | S014_When_Does_a_Sequence_Converge.txt | So we looked at the formal definition of what it means for a sequence to converge, but as a practical matter, how can we tell whether a given sequence converges or not? There are two criteria that are the most commonly used for that purpose, and it's useful to be aware of them. The first one deals with the case where w... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L047_Partitions.txt | We now come to our last major class of counting problems. We will count the number of ways that a given set can be partitioned into pieces of given sizes. We start with a set that consists of n different elements. And we have r persons. We want to give n1 items to the first person, give n2 items to the second person, a... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L035_Conditional_Independence.txt | Conditional probabilities are like ordinary probabilities, except that they apply to a new situation where some additional information is available. For this reason, any concept relevant to probability models has a counterpart that applies to conditional probability models. In this spirit, we can define a notion of con... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L064_Conditional_PMFs_Expectations_Given_an_Event.txt | We now move to a new topic-- conditioning. Every probabilistic concept or probabilistic fact has a conditional counterpart. As we have seen before, we can start with a probabilistic model and some initial probabilities. But then if we are told that the certain event has occurred, we can revise our model and consider co... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L053_Probability_Mass_Functions.txt | A random variable can take different numerical values depending on the outcome of the experiment. Some outcomes are more likely than others, and similarly some of the possible numerical values of a random variable will be more likely than others. We restrict ourselves to discrete random variables, and we will describe ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L023_A_Die_Roll_Example.txt | This is a simple example where we want to just apply the formula for conditional probabilities and see what we get. The example involves a four-sided die, if you can imagine such an object, which we roll twice, and we record the first roll, and the second roll. So there are 16 possible outcomes. We assume to keep thing... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L086_Exponential_Random_Variables.txt | We now introduce a new of random variable, the exponential random variable. It has a probability density function that is determined by a single parameter lambda, which is a positive number. And the form of the PDF is as shown here. Note that the PDF is equal to 0 when x is negative, which means that negative values of... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L055_Uniform_Random_Variables.txt | In this segment and the next two, we will introduce a few useful random variables that show up in many applications-- discrete uniform random variables, binomial random variables, and geometric random variables So let's start with a discrete uniform. A discrete uniform random variable is one that has a PMF of this form... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L046_A_Coin_Tossing_Example.txt | Let us now put to use our understanding of the coin-tossing model and the associated binomial probabilities. We will solve the following problem. We have a coin, which is tossed 10 times. And we're told that exactly three out of the 10 tosses resulted in heads. Given this information, we would like to calculate the pro... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L049_Multinomial_Probabilities.txt | PROFESSOR: In this segment, we will discuss the multinomial model and the multinomial probabilities, which are a nice generalization of the binomial probabilities. The setting is as follows. We are dealing with balls and the balls come into different colors. There are r possible different colors. We pick a ball at rand... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L034_Independence_of_Event_Complements.txt | Let us now discuss an interesting fact about independence that should enhance our understanding. Suppose that events A and B are independent. Intuitively, if I tell you that A occurred, this does not change your beliefs as to the likelihood that B will occur. But in that case, this should not change your beliefs as to ... |
MIT_RES6012_Introduction_to_Probability_Spring_2018 | L082_Probability_Density_Functions.txt | In this segment, we introduce the concept of continuous random variables and their characterization in terms of probability density functions, or PDFs for short. Let us first go back to discrete random variables. A discrete random variable takes values in a discrete set. There is a total of one unit of probability assi... |
US_History | Womens_Suffrage_Crash_Course_US_History_31.txt | Episode 31: Feminism and Suffrage Hi, I’m John Green, this is Crash Course U.S. history and today we’re going to talk about women in the progressive era. My God, that is a fantastic hat. Wait, votes for women?? So between Teddy Roosevelt, and Woodrow Wilson, and all those doughboys headed off to war, women in this peri... |
US_History | Thomas_Jefferson_His_Democracy_Crash_Course_US_History_10.txt | Hi I'm John Green, this is Crash Course US History, and today we're going to discuss Thomas Jefferson. We're going to learn about how America became a thriving nation of small, independent farmers, eschewing manufacturing and world trade, and becoming the richest and most powerful nation in the world in the 19th centur... |
US_History | Where_US_Politics_Came_From_Crash_Course_US_History_9.txt | Hi, I'm John Green, and this is Crash Course U.S. History, and now that we have a Constitution, it’s actually United States history. Today we’re going to look at the birth of America’s pastime. No, Stan, not baseball. Not football. Not eating. I mean politics, which in America has been adversarial since its very beginn... |
US_History | World_War_II_Part_1_Crash_Course_US_History_35.txt | Hi, I’m John Green, this is Crash Course U.S. history, and today we’re going to talk about a topic so huge to history buffs that we can only discuss a tiny, little fraction of it. I am of course referring to paratroopering. No World War II. World War II is the only historical event that has, like, its own cable channel... |
US_History | War_Expansion_Crash_Course_US_History_17.txt | Hi, I'm John Green, this is Crash Course U.S. history and today we’re going to discuss how the United States came to acquire two of its largest states, Texas and…there is another one. Mr. Green! Mr. Green! I believe the answer you’re looking for is Alaska. Oh me from the past, as you can clearly tell from the globe, Al... |
US_History | The_Clinton_Years_or_the_1990s_Crash_Course_US_History_45.txt | Hi, I’m John Green, this is CrashCourse U.S. history, and today we have finally reached the Clinton years. Bill Clinton and I are really quite similar, actually. We were both brought up in the South. We both come from broken families … well, no, not actually. Also, I did not attended any Ivy League University. Yeah, I’... |
US_History | Civil_Rights_and_the_1950s_Crash_Course_US_History_39.txt | Episode 39: Consensus and Protest: Civil Rights LOCKED Hi, I’m John Green, this is Crash Course U.S. history and today we’re going to look at one of the most important periods of American social history, the 1950s. Why is it so important? Well, first because it saw the advent of the greatest invention in human history:... |
US_History | Reconstruction_and_1876_Crash_Course_US_History_22.txt | Episode 21: Reconstruction Hi, I’m John Green, this is Crash Course U.S. History and huzzah! The Civil War is over! The slaves are free! Huzzah! That one hit me in the head? It’s very dangerous, Crash Course. So when you say, “Don’t aim at a person,” that includes myself? The roller coaster only goes up from here, my f... |
US_History | Obamanation_Crash_Course_US_History_47.txt | Hi, I’m John Green, this is CrashCourse U.S. history and we’ve finally done it, we’ve reached the end of history! Ahh. Stan, says that history never ends but whatever we’ve reached the part that Me From the Present is in. So we aren’t going to cover the astonishing results of the 2016 presidential election. What we are... |
US_History | The_War_of_1812_Crash_Course_US_History_11.txt | Hi, I'm John Green, this is Crash Course US History and today we're going to talk about what America's best at: War. [Patriotic Rock Music] Uh, Mr. Green, the United States has actually only declared war 5 times in the last 230 years. Oh me from the the past, you sniveling literalist. Well today we're going to talk abo... |
US_History | The_1960s_in_America_Crash_Course_US_History_40.txt | Hi, I’m John Green, this is Crash Course US History and today we’re gonna talk about the 1960s. Mr. Green, Mr. Green. Great. The decade made famous by the narcissists who lived through it. Hey, Me From the Past, finally you and I agree about something wholeheartedly. But while I don’t wish to indulge the baby-boomers’ ... |
US_History | Crash_Course_US_History_Preview.txt | hi and welcome to crash course u.s history a course that approximately follows the ap u.s history curriculum as it appeared in 2013 and even though we call it u.s history we do begin before any europeans show up in north america from the 15th century we'll take you through every beat of the country's history until we r... |
US_History | The_Rise_of_Conservatism_Crash_Course_US_History_41.txt | Episode 41: Rise of Conservatism Hi, I’m John Green, this is CrashCourse U.S. history and today we’re going to--Nixon?--we’re going to talk about the rise of conservatism. So Alabama, where I went to high school, is a pretty conservative state and reliably sends Republicans to Washington. Like, both of its Senators, Je... |
US_History | The_Seven_Years_War_and_the_Great_Awakening_Crash_Course_US_History_5.txt | Hi, I'm John Green. This is Crash Course U.S. History. And today we're going to discuss the events that led to the events that led to the American revolution. So, we'll begin with the Seven Years War which, as Crash Course World History fans will remember, Winston Churchill referred to as as the "First World War". The ... |
US_History | Gilded_Age_Politics_Crash_Course_US_History_26.txt | Hi, I’m John Green, this is Crash Course: US History, and today we’re going to continue our look at the Gilded Age by focusing on political science. Mr. Green, Mr. Green, so it’s another history class where we don’t actually talk about history? Oh, Me From the Past, your insistence on trying to place academic explorati... |
US_History | The_Reagan_Revolution_Crash_Course_US_History_43.txt | Hi, I'm John Green, this is Crash Course U.S. history, and today we're going to talk about the guy who arguably did the most to shape the world that I live in. NO, Stan not Carrottop. No, not Cumberbatch although he did do the most to shape the Tumblr that I live in. I'm talking about The Great Communicator: Ronald Rea... |
US_History | World_War_II_Part_2_The_Homefront_Crash_Course_US_History_36.txt | Episode 36: World War II (2) – the war at home Hi, I’m John Green, this is Crash Course U.S. History and today we’re going to discuss how World War II played out at home and also the meaning of the war. Mr. Green, Mr. Green, so is this going to be, like, one of the boring philosophical ones, then? Oh, Me From the Past,... |
US_History | The_Progressive_Era_Crash_Course_US_History_27.txt | Episode 27: Progressive Era Hi, I’m John Green, this is CrashCourse U.S. history, and today we’re gonna talk about Progressives. No Stan Progressives. Yes. You know, like these guys who used to want to bomb the means of production, but also less radical Progressives. Mr. Green, Mr. Green. Are we talking about, like, tu... |
US_History | The_New_Deal_Crash_Course_US_History_34.txt | Episode 34 – The New Deal Hi, I’m John Green, this is CrashCourse U.S. history, and today we’re going to get a little bit controversial, as we discuss the FDR administration’s response to the Great Depression: the New Deal. That’s the National Recovery Administration, by the way, not the National Rifle Association or t... |
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