Stat 371 1St Edition Exam 2 Study Guide Lectures 8 12 Lecture 8 September 22 Random Variable that takes on numerical values depending on the outcome of an experiment o Represented with a capital letter represents every outcome an experiment can have Discrete Random variables just like discrete variable a discrete random variable can 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 value o y yi f yi f yi probability of it occurring Variance of Random Variable o 2y Var Y E Y2 E Y 2 o E Y2 yi2 f yi o Standard deviation Var Y o Multiply adding by a constant E cX cE X Var cX c2Var X E aX b aE X b Var aX b a2Var X o Adding random variables E z E x E y Var z Var x Var y Okay as long as x and y are independent o Subtracting 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 distributions o Binomial distributions o Poison distribution Continuous probability distributions o Exponential distribution o Uniform distribution Binomial distribution a discrete probability distribution that consist of n independent trials with each trial having exactly two outcomes o Properties 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 o Properties 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 distributions o P x X nx px 1 p n x o 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 pdf o 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 x o 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 a particular 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 1 o BE SURE TO ADJUST THE TIME FRAME WHEN X o 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 under every continuous graph must equal 1 o impossible to take the area of line o 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 x o Variance of Exponential distribution 2 Uniform Distribution characterized by each point having equal probability of occurring o Probability is the area of interest height x width o Height 1b a o mean of uniform distribution a b2 o variance of uniform distribution b a 212 Lecture 12 October 1 Properties of Normal Distribution aka bell curve 3 unique normal distribution for each value of of the mode of distribution occurs at which is also the median distribution is symmetric with the tails extending to infinity determines width of curve area under curve 1 Empirical rule o 68 of data within 1 SD o 95 of data within 2 SD o 99 of data within 3 SD Standard normal distribution 0 1 aprx integral of standard normal distribution 616 617 z standard normal problem Convert numbers standardizing equation o when we have a normal distribution w mean and standard deviation we have to convert it to make it have a mean of 0 and SD of 1 o To make mean 0 subtract off the mean from random variable 0 o 1 divide by SD 0 0 1 o z x z represents the of SD s that x is from the mean How to use the table o z values represented down first column and across first row o careful of positive and negative z values o numbers inside the table represent area TO THE LEFT of the z value o to find area to the right P z 1 P z P z x 0 Lecture 13 October 3 more notation about z z 025 P Z z 025 o this represents the AREA TO THE RIGHT Lecture 14 October 6 Review Lecture 15 October 8 Sampling Distribution Sampling Distribution we take a sample size n we are interested in finding the distribution of the sample mean Yor X want to know the mean SD o Standardizing formula z X n o Mean x o SD n o Whenever you see sample size use n Central Limit Theorem as n …
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