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X~ G(p) means that the discrete random variableXhas a geometric probability distribution with probability of success in a single trialp. | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
X= the number of independent trials until the first success | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
Xtakes on the valuesx= 1, 2, 3, ... | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
p= the probability of a success for any trial | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
q= the probability of a failure for any trialp+q= 1q= 1 âp | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
The mean isμ=1p1p. | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
The standard deviation isÏ=1âpp21âpp2=1p(1pâ1)1p(1pâ1). | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
X~H(r,b,n) means that the discrete random variableXhas a hypergeometric probability distribution withr= the size of the group of interest (first group),b= the size of the second group, andn= the size of the chosen sample. | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
X= the number of items from the group of interest that are in the chosen sample, andXmay take on the valuesx= 0, 1, ..., up to the size of the group of interest. (The minimum value forXmay be larger than zero in some instances.) | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
nâ¤r+b | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
The mean ofXis given by the formulaμ=nrr+bnrr+band the standard deviation is =rbn(r+bân)(r+b)2(r+bâ1)rbn(r+bân)(r+b)2(r+bâ1). | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
X~P(μ) means thatXhas a Poisson probability distribution whereX= the number of occurrences in the interval of interest. | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
Xtakes on the valuesx= 0, 1, 2, 3, ... | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
The meanμis typically given. | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
The variance isÏ2=μ, and the standard deviation isÏ=μÏ=μ. | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
The probability of having exactlyxxsuccesses inrrtrials isPX=x=e-μμxx!PX=x=e-μμxx!. | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
WhenP(μ) is used to approximate a binomial distribution,μ=npwherenrepresents the number of independent trials andprepresents the probability of success in a single trial. | https://openstax.org/books/introductory-statistics-2e/pages/4-formula-review |
Bernoulli Trials : an experiment with the following characteristics:There are only two possible outcomes called âsuccessâ and âfailureâ for each trial.The probabilitypof a success is the same for any trial (so the probabilityq= 1 âpof a failure is the same for any trial). | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
Binomial Experiment : a statistical experiment that satisfies the following three conditions:There are a fixed number of trials,n.There are only two possible outcomes, called "success" and, "failure," for each trial. The letterpdenotes the probability of a success on one trial, andqdenotes the probability of a failure ... | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
Binomial Probability Distribution : a discrete random variable (RV) that arises from Bernoulli trials; there are a fixed number,n, of independent trials. âIndependentâ means that the result of any trial (for example, trial one) does not affect the results of the following trials, and all trials are conducted under ... | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
Expected Value : expected arithmetic average when an experiment is repeated many times; also called the mean. Notations:μ. For a discrete random variable (RV) with probability distribution functionP(x),the definition can also be written in the formμ=ââxP(x). | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
Geometric Distribution : a discrete random variable (RV) that arises from the Bernoulli trials; the trials are repeated until the first success. The geometric variableXis defined as the number of trials until the first success. Notation:X~G(p). The mean isμ=1p1pand the standard deviation isÏ=1p(1pâ1)1p(1pâ1). The... | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
Geometric Experiment : a statistical experiment with the following properties:There are one or more Bernoulli trials with all failures except the last one, which is a success.In theory, the number of trials could go on forever. There must be at least one trial.The probability,p, of a success and the probability,q, of a... | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
Hypergeometric Experiment : a statistical experiment with the following properties:You take samples from two groups.You are concerned with a group of interest, called the first group.You sample without replacement from the combined groups.Each pick is not independent, since sampling is without replacement.You are not d... | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
Hypergeometric Probability : a discrete random variable (RV) that is characterized by:A fixed number of trials.The probability of success is not the same from trial to trial.We sample from two groups of items when we are interested in only one group.Xis defined as the number of successes out of the total number of item... | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
Mean : a number that measures the central tendency; a common name for mean is âaverage.â The term âmeanâ is a shortened form of âarithmetic mean.â By definition, the mean for a sample (detonated byx¯x¯) isx¯=SumofallvaluesinthesampleNumberofvaluesinthesamplex¯=SumofallvaluesinthesampleNumberofvaluesinth... | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
Mean of a Probability Distribution : the long-term average of many trials of a statistical experiment | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
Poisson Probability Distribution : a discrete random variable (RV) that counts the number of times a certain event will occur in a specific interval; characteristics of the variable:The probability that the event occurs in a given interval is the same for all intervals.The events occur with a known mean and independent... | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
Probability Distribution Function (PDF) : a mathematical description of a discrete random variable (RV), given either in the form of an equation (formula) or in the form of a table listing all the possible outcomes of an experiment and the probability associated with each outcome. | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
Random Variable (RV) : a characteristic of interest in a population being studied; common notation for variables are upper case Latin lettersX,Y,Z,...; common notation for a specific value from the domain (set of all possible values of a variable) are lower case Latin lettersx, y,andz. For example, ifXis the number of ... | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
Standard Deviation of a Probability Distribution : a number that measures how far the outcomes of a statistical experiment are from the mean of the distributionÏ=â[xâμ2âΡx]Ï=â[xâμ2âΡx] | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
The Law of Large Numbers : As the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency probability approaches zero. | https://openstax.org/books/introductory-statistics-2e/pages/4-key-terms |
The probability density function (pdf) is used to describe probabilities for continuous random variables. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. In other words, the area under the density curve between pointsaandbis equal toP(... | https://openstax.org/books/introductory-statistics-2e/pages/5-chapter-review |
The cumulative distribution function (cdf) ofXis defined byP(Xâ¤x). It is a function ofxthat gives the probability that the random variable is less than or equal tox. | https://openstax.org/books/introductory-statistics-2e/pages/5-chapter-review |
IfXhas a uniform distribution wherea<x<boraâ¤xâ¤b, thenXtakes on values betweenaandb(may includeaandb). All valuesxare equally likely. We writeXâ¼U(a,b). The mean ofXisμ=a+b2μ=a+b2. The standard deviation ofXisÏ=(bâa)212Ï=(bâa)212. The probability density function ofXisf(x)=1bâaf(x)=1bâaforaâ¤xâ¤b. The... | https://openstax.org/books/introductory-statistics-2e/pages/5-chapter-review |
The probabilityP(c<X<d) may be found by computing the area underf(x), betweencandd. Since the corresponding area is a rectangle, the area may be found simply by multiplying the width and the height. | https://openstax.org/books/introductory-statistics-2e/pages/5-chapter-review |
IfXhas anexponential distributionwith meanμ, then thedecay parameterism=1μ1μ, and we writeXâ¼Exp(m) wherex⥠0 andm> 0 . The probability density function ofXisf(x) =me-mx(or equivalentlyf(x)=1μeâx/μf(x)=1μeâx/μ. The cumulative distribution function ofXisP(Xâ¤x) = 1 âeâmx. | https://openstax.org/books/introductory-statistics-2e/pages/5-chapter-review |
The exponential distribution has thememoryless property, which says that future probabilities do not depend on any past information. Mathematically, it says thatP(X>x+k|X>x) =P(X>k). | https://openstax.org/books/introductory-statistics-2e/pages/5-chapter-review |
IfTrepresents the waiting time between events, and ifTâ¼Exp(λ), then the number of eventsXper unit time follows the Poisson distribution with meanλ. The probability density function ofXisP(X=k)=λkeâkk!P(X=k)=λkeâkk!. This may be computed using a TI-83, 83+, 84, 84+ calculator with the command poissonpdf(λ,k).... | https://openstax.org/books/introductory-statistics-2e/pages/5-chapter-review |
Probability density function (pdf)f(x): | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
f(x) ⥠0 | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
The total area under the curvef(x) is one. | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
Cumulative distribution function (cdf):P(Xâ¤x) | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
X= a real number betweenaandb(in some instances,Xcan take on the valuesaandb).a= smallestX;b= largestX | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
X~U(a, b) | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
The mean isμ=a+b2μ=a+b2 | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
The standard deviation isÏ=(bâa)212Ï=(bâa)212 | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
Probability density function:f(x)=1bâaf(x)=1bâaforaâ¤Xâ¤baâ¤Xâ¤b | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
Area to the Left ofx:P(X<x) = (xâa)(1bâa)(1bâa) | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
Area to the Right ofx:P(X>x) = (bâx)(1bâa)(1bâa) | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
Area Betweencandd:P(c<x<d) = (base)(height) = (dâc)(1bâa)(1bâa) | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
Uniform:X~U(a,b) wherea<x<b | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
pdf:f(x)=1bâaf(x)=1bâafora ⤠x ⤠b | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
cdf:P(Xâ¤x) =xâabâaxâabâa | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
meanµ=a+b2a+b2 | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
standard deviationÏ=(bâa)212=(bâa)212 | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
P(c<X<d) = (dâc)(1bâa)(1bâa) | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
Exponential:X~Exp(m) wherem= the decay parameter | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
pdf:f(x) =me(âmx)wherex⥠0 andm> 0 | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
cdf:P(Xâ¤x) = 1 âe(âmx) | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
meanµ=1m1m | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
standard deviationÏ=µ | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
percentilek:k=ln(1âAreaToTheLeftOfk)(âm)ln(1âAreaToTheLeftOfk)(âm) | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
AdditionallyP(X>x) =e(âmx)P(a<X<b) =e(âma)âe(âmb) | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
P(X>x) =e(âmx) | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
P(a<X<b) =e(âma)âe(âmb) | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
Memoryless Property:P(X>x+k|X>x) =P(X>k) | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
Poisson probability:P(X=k)=λkeâkk!P(X=k)=λkeâkk!with meanλ | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
k! =k*(k-1)*(k-2)*(k-3)*â¦3*2*1 | https://openstax.org/books/introductory-statistics-2e/pages/5-formula-review |
Conditional Probability : the likelihood that an event will occur given that another event has already occurred. | https://openstax.org/books/introductory-statistics-2e/pages/5-key-terms |
decay parameter : The decay parameter describes the rate at which probabilities decay to zero for increasing values ofx. It is the valuemin the probability density functionf(x) =me(-mx)of an exponential random variable. It is also equal tom=1μ1μ, whereμis the mean of the random variable. | https://openstax.org/books/introductory-statistics-2e/pages/5-key-terms |
Exponential Distribution : a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital; the notation isX~Exp(m). The mean isμ=1m1mand the standard deviation is Ï =1m1m. The probabili... | https://openstax.org/books/introductory-statistics-2e/pages/5-key-terms |
memoryless property : For an exponential random variableX, the memoryless property is the statement that knowledge of what has occurred in the past has no effect on future probabilities. This means that the probability thatXexceedsx+k, given that it has exceededx, is the same as the probability thatXwould exceedkif we ... | https://openstax.org/books/introductory-statistics-2e/pages/5-key-terms |
Poisson distribution : If there is a known average ofλevents occurring per unit time, and these events are independent of each other, then the number of eventsXoccurring in one unit of time has the Poisson distribution. The probability ofkevents occurring in one unit time is equal toP(X=k)=λkeâλk!P(X=k)=λkeâλk... | https://openstax.org/books/introductory-statistics-2e/pages/5-key-terms |
Uniform Distribution : a continuous random variable (RV) that has equally likely outcomes over the domain,a<x<b. Notation:X~U(a,b). The mean isμ=a+b2a+b2and the standard deviation isÏ=(bâa)212Ï=(bâa)212. The probability density function isf(x) =1bâa1bâafora<x<boraâ¤xâ¤b. The cumulative distribution isP(Xâ... | https://openstax.org/books/introductory-statistics-2e/pages/5-key-terms |
Az-score is a standardized value. Its distribution is the standard normal,Z~N(0, 1). The mean of thez-scores is zero and the standard deviation is one. Ifzis thez-score for a valuexfrom the normal distributionN(µ,Ï) thenztells you how many standard deviationsxis above (greater than) or below (less than)µ. | https://openstax.org/books/introductory-statistics-2e/pages/6-chapter-review |
The normal distribution, which is continuous, is the most important of all the probability distributions. Its graph is bell-shaped. This bell-shaped curve is used in almost all disciplines. Since it is a continuous distribution, the total area under the curve is one. The parameters of the normal are the meanµand the ... | https://openstax.org/books/introductory-statistics-2e/pages/6-chapter-review |
Xâ¼N(μ,Ï) | https://openstax.org/books/introductory-statistics-2e/pages/6-formula-review |
μ= the mean;Ï= the standard deviation | https://openstax.org/books/introductory-statistics-2e/pages/6-formula-review |
z= a standardized value (z-score) | https://openstax.org/books/introductory-statistics-2e/pages/6-formula-review |
mean = 0; standard deviation = 1 | https://openstax.org/books/introductory-statistics-2e/pages/6-formula-review |
To find the observed value,x, when thez-scores is known:x=μ+ (z)Ï | https://openstax.org/books/introductory-statistics-2e/pages/6-formula-review |
z-score:z=xâμÏxâÎ¼Ï | https://openstax.org/books/introductory-statistics-2e/pages/6-formula-review |
Z= the random variable forz-scores | https://openstax.org/books/introductory-statistics-2e/pages/6-formula-review |
Normal Distribution:X~N(µ,Ï) whereµis the mean andÏis the standard deviation. | https://openstax.org/books/introductory-statistics-2e/pages/6-formula-review |
Standard Normal Distribution:Z~N(0, 1). | https://openstax.org/books/introductory-statistics-2e/pages/6-formula-review |
Calculator function for probability: normalcdf (lowerxvalue of the area, upperxvalue of the area, mean, standard deviation) | https://openstax.org/books/introductory-statistics-2e/pages/6-formula-review |
Calculator function for thekthpercentile:k= invNorm (area to the left ofk, mean, standard deviation) | https://openstax.org/books/introductory-statistics-2e/pages/6-formula-review |
Normal Distribution : a continuous random variable (RV) with pdff(x) =1Ï2Ïeâ(xâμ)2Ï221Ï2Ïeâ(xâμ)2Ï22, whereμis the mean of the distribution andÏis the standard deviation; notation:X~N(μ,Ï). Ifμ= 0 andÏ= 1, the RV is called thestandard normal distribution. | https://openstax.org/books/introductory-statistics-2e/pages/6-key-terms |
Standard Normal Distribution : a continuous random variable (RV)X~N(0, 1); whenXfollows the standard normal distribution, it is often noted asZ~N(0, 1). | https://openstax.org/books/introductory-statistics-2e/pages/6-key-terms |
z-score : the linear transformation of the formz=xâμÏxâμÏ; if this transformation is applied to any normal distributionX~N(μ,Ï) the result is the standard normal distributionZ~N(0,1). If this transformation is applied to any specific valuexof the RV with meanμand standard deviationÏ, the result is called th... | https://openstax.org/books/introductory-statistics-2e/pages/6-key-terms |
In a population whose distribution may be known or unknown, if the size (n) of samples is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the sample means will equal the population mean. The standard deviation of the distribution of the sample means, called the standar... | https://openstax.org/books/introductory-statistics-2e/pages/7-chapter-review |
The central limit theorem tells us that for a population with any distribution, the distribution of the sums for the sample means approaches a normal distribution as the sample size increases. In other words, if the sample size is large enough, the distribution of the sums can be approximated by a normal distribution e... | https://openstax.org/books/introductory-statistics-2e/pages/7-chapter-review |
The central limit theorem can be used to illustrate the law of large numbers. The law of large numbers states that the larger the sample size you take from a population, the closer the sample meanx¯x¯gets toμ. | https://openstax.org/books/introductory-statistics-2e/pages/7-chapter-review |
The Central Limit Theorem for Sample Means:x¯x¯~N(μx,Ïxn)(μx,Ïxn) | https://openstax.org/books/introductory-statistics-2e/pages/7-formula-review |
The MeanX¯X¯:μx | https://openstax.org/books/introductory-statistics-2e/pages/7-formula-review |
Central Limit Theorem for Sample Means z-score and standard error of the mean:z=x¯âμx(Ïxn)z=x¯âμx(Ïxn) | https://openstax.org/books/introductory-statistics-2e/pages/7-formula-review |
Standard Error of the Mean (Standard Deviation (X¯X¯)):ÏxnÏxn | https://openstax.org/books/introductory-statistics-2e/pages/7-formula-review |
The Central Limit Theorem for Sums:âX~N[(n)(μx),(nn)(Ïx)] | https://openstax.org/books/introductory-statistics-2e/pages/7-formula-review |
Mean for Sums (âX): (n)(μx) | https://openstax.org/books/introductory-statistics-2e/pages/7-formula-review |
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