Stat 371 1St EditionExam # 2 Study Guide Lectures: 8 - 12Lecture 8 (September 22)- Random Variable: that takes on numerical values depending on the outcome of an experimentoRepresented with a capital letter; represents every outcome an experiment can have- Discrete Random variables: just like discrete variable, a discrete random variablecan only take values of specific numbers- Discrete Probability Distribution: a table, graph, or equation that describes the values of random variable and its associated probabilities- Entire f(x) column must add up to 1- Mean of discrete probability distribution=Expected valueo-y=-yi * f(yi) [f(yi): probability of it occurring)- Variance of Random Variable:o-2y=Var(Y)=E(Y2)-[E(Y)]2oE(Y2)=- (yi2)(f(yi))oStandard deviation: -Var(Y)oMultiply/adding by a constant: E(cX)=cE(X) Var(cX)=c2Var(X) E(aX+b)=aE(X)+b Var(aX+b)=a2Var(X)oAdding random variables: E(z)=E(x)+E(y) Var(z)=Var(x)+Var(y)- Okay as long as x and y are independentoSubtracting random variables: E(z)=E(x)-E(y) Var(z)=Var(x)+Var(y)- Still add variances, can only do this with variances not standard deviation- Discrete probability distributionsoBinomial distributionsoPoison distribution- Continuous probability distributionsoExponential distributionoUniform distribution- Binomial distribution: a discrete probability distribution that consist of n independent trials with each trial having exactly two outcomes oProperties: Overall experiment has n identical, independent trials Each trial has 2 possible outcomes (fail or succeed) P(success)=p; P(failure)=1-p The binomial random variable, X, represents the count of the number of successes. X can take values 0 to n. Lecture 9 (September 24): Mean and variance of Random variables- Binomial distribution: a discrete probability distribution that consist of n independent trials with each trial having exactly two outcomes oProperties: Overall experiment has n identical, independent trials Each trial has 2 possible outcomes (fail or succeed) P(success)=p; P(failure)=1-p The binomial random variable, X, represents the count of the number of successes. X can take values 0 to n. - Binomial distributionso P(x=X): (nx)px(1-p)n-xo (nx)=n!x!(n-x)!o Calculator: use formula nCr (n is our n, r is x) “n” on home screen → “MATH” → “PRB” → option 3: nCr→ enter x value → “ENTER”o P(x=#) ⇒ pdfo P(x>#), P(x<#) ⇒ cumlative density function (cdf) i=1npdf(i)Lecture 10 (September 26)- pdf: P(X=x)=(nx)px(1-p)n-x- Mean of the binomial distribution: E(x)=np- Variance: Var(x)=np(1-p)o denote binomial distributions like: Bin (np, np(1-p))o therefore if you see Bin(5, 1.5), this means a binomial of mean 5 and variance of 1.5- Calculator for finding P(X=x)o 2nd VARS (DISTSR) → option a: binompdf( → enter: binompdf(n,p,x) → gives probability that X=xo If finding cdf: gives probability of P(Xx) if need to find P(x<2); you would need to type in binomcdf(n,p,1) as P(x<2)=P(x1)2 if need to find P(x>4), you will need to type in 1-binomcdf(n,p,4) as P(x>4) + P(x4)= 1 - Poisson Distribution: used when we are dealing with a number of occurrence of aparticular event over a specified interval of time or space. Number of people that go into Starbucks in an hour.- Assumptions:o Probability of an occurrence of the event is the same for any 2 intervals of equal length. Probability of getting a customer in the first hour is equal to the probability of getting a customer in the second hour. o The occurrence of the event in any interval is independent of the occurrence in any other interval. If a customer comes in the first hour, it has no effect on the customer coming in the second hour.o P(x=X)=e-⋋⋋xx! ⋋=expected value of events to occur in time interval t (average) x=0,1,2,3,4 (every integer, time for x-∞( 0!=1o BE SURE TO ADJUST THE TIME FRAME WHEN ⋋≠Xo Mean AND variance of Poisson= ⋋= Expected value (mean)o On calculator pdf (x=#): 2nd VARS (distr) → option C: poissonpd( → enter in poisson pdf (⋋,x) cdf (X><#): option D:poissoncdf( → poisson cdf(⋋,x) → this gives P(x#)Lecture 11 (September 29): Continuous: Exponential Distribution- Continuous Distributions: described by graphs and equations where the area under the curve of the graph represents the probability; therefore, the areal underevery continuous graph must equal 1o impossible to take the area of lineo inverse of poisson’s distribution Poisson: events/time Continuous Distribution: time between events- Exponential Distribution: use cdf (integral of pdf)o P(xX)=1-e-x/⋋o if wanted P(x>X), just use compliment 1-CDF since dealing with time, x must be positive.o P(axb)=P(x<b)-P(x<a)o Mean of exponential distribution: xo Variance of Exponential distribution: ⋋2- Uniform Distribution: characterized by each point having equal probability of occurringo Probability is the area of interest (height x width)o Height: 1b-ao mean of uniform distribution: a+b2o variance of uniform distribution: (b-a)212Lecture 12 (October 1)- Properties of Normal Distribution: aka bell curve3o unique normal distribution for each value of μ of σo the mode of distribution occurs at μ, which is also the mediano distribution is symmetric with the tails extending to infinityo σ determines width of curveo area under curve =1- Empirical rule:o 68% of data within 1 SDo 95% of data within 2 SDo 99% of data within 3 SD- Standard normal distribution → μ=0, σ=1, aprx. integral of standard normal distribution 616-617, z= standard normal problem - Convert numbers: standardizing equationo when we have a normal distribution w/ mean μ and standard deviation σ, we have to convert it to make it have a mean of 0 and SD of 1o To make mean=0, subtract off the mean from random variable (μ-μ=0)o σ=1 → divide by SD (0/0 = 1)o z=x-μσ⇒ z represents the # of SD’s that x is from the mean- How to use the tableo z values represented down first column and across first rowo careful of positive and negative z valueso numbers inside the table represent area TO THE LEFT of the z valueo to find area to the right P(z) → 1-P(z); P(z=x)=0Lecture 13 (October 3)- more notation about z → z.025=P(Z>z) =.025o this represents the AREA TO THE RIGHTLecture 14 (October 6): ReviewLecture 15 (October 8): Sampling Distribution- Sampling Distribution: we take a sample size n, we
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