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# PROBABILITY STAT430

Penn

GPA 3.53

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This 20 page Class Notes was uploaded by Orval Funk on Monday September 28, 2015. The Class Notes belongs to STAT430 at University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 28 views. For similar materials see /class/215433/stat430-university-of-pennsylvania in Statistics at University of Pennsylvania.

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Date Created: 09/28/15

STAT 430510 Probability Hui Nie Lecture 7 June 4th 2009 Review Properties of Probability o Conditional Probability o The Law of Total Probability o Bayes Formula 0 Independence Random Variables o In most problems we are interested only in a particular aspect of the outcomes of experiments 0 Example When we toss 10 coins we are interested in the total number of heads and not the outcome for each coin Definition o For a given sample space S a random variable rv is a realvalued function defined over the elements of S Example 0 Suppose that our experiment consists of tossing 3 fair coins If we let Y denote the number of heads that appear then Y is a random variable taking on one of the values 012 and 3 with respective probabilities PY0 PTTT PY1 PTTHTHTHTT PY2 PTHHHTHHHT PY 3 PH H H g Example 0 Three balls are to be randomly selected without replacement from an urn containing 20 balls numbered 1 through 20 If we bet that at least one of the balls that are drawn has a number no less than 17 what is the probability that we win the bet Random Variables Continued 0 A random variable reflects the aspect of a random experiment that is of interest to us 0 There are two types of random variables 0 A discrete random variable has at most a countable number of possible values 0 A continuous random variable takes all values in an interval of numbers Example of Discrete Random Variables o Xthe sum of two tosses of a fair die Possible values 234 12 o Xthe total number of coin tosses required for three consecutive heads X is a discrete random variable The set of possible values is infinite but countable Probability Mass Function 0 If X is a discrete random variable the function given by pX PX X Pall s e S Xs X for each x within the range of X is called the probability mass function of X Cumulative Distribution Function 0 The cumulative distribution function FX of a discrete random variable X with pml pX is given by FX PX S X 2100 VSX o For any X FX is the probability that the observed value of X will be at most X Example Again 0 Suppose that our experiment consists of tossing 3 fair coins If we let Y denote the number of heads that appear then Y is a random variable taking on one of the values 012 and 3 with respective probabilities proPYo p1PY1 przPY2 prsPY3 PT T T PT T H T H T H T T PT H H H T H H H T PH H H coicooolw Example Again Continued FO PY O F1 PY 1 F2 PY g 2 F3 PY g 3 COllgt CO Expected Value 0 If X is a discrete random variable having a probability mass function px then the expected value of X is defined by Em Z W xpxgt0 Expected Value Continued 0 EX is a weighted average of the possible values that X can take on 0 Each value is weighted by the probability that X takes on it Example Continued 0 Suppose that our experiment consists of tossing 3 fair coins If we let Y denote the number of heads that appear What is E Y o EYOxp01xp12xp23xp3g Example 0 Find EX where X is the outcome when we roll a fair die 0 Solution 35 But why Expectation of a Function of a Random Variable o If X is a discrete random variable that takes on one of the values X i 2 1 with respective probabilities p then for any realvalued function g E 9X 9XIPXI Example 0 Let X denote a random variable that takes on any of the values 1 O and 1 with respective probabilities PX 71 02 PX O 05 PX 1 03 Compute EX2 0 Solution EX2 712PX 71 02PX o12PX 1 05 Corollary 0 If a and b are constant then EaX b aEX b Example 0 Assume that EX 1 Calculate E7X 2 0 Solution E7X 2 7EX 2 9

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