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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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