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Math 160 Intro to Statistics OCC Section 5.2 – The Mean and Standard Deviation of a Discrete Random Variable Mean of a Discrete Random Variable The mean of a discrete random variable X is denoted by X or, when no confusion will arise, simply . It is defined by xP ( X x) or xP( x) The terms expected value and expectation are commonly used in place of the term mean. Interpretation of the Mean of a Random Variable In a large number of independent observations of a random variable X, the average value of those observations will approximate equal the mean, , of X; The larger the number of observations, the closer the average tends to be . Standard Deviation of a Discrete Random Variable The standard deviation of a discrete random variable X is denoted by X or, when no confusion will arise, simply . It is defined as ( x ) 2 P( X x) or ( x )2 P( x) The standard deviation of a discrete random variable X can also be obtained from the computing formula x 2 P( X x) 2 or x 2 P( x) 2 Exercises: 1) What is the mean number of siblings for Exercise #1 from the handout in Section 5.1? x P(X = x) x·P(X = x) -1- Math 160 Intro to Statistics OCC Section 5.2 – The Mean and Standard Deviation of a Discrete Random Variable Exercises: 2) What is the standard deviation for the number of siblings for Exercise #1 from the handout in Section 5.1? x2 x 3) P(X = x) x2·P(X = x) Roulette is a common game of gambling found in Casinos. An American roulette wheel contains 38 numbers: 18 are red, 18 are black, and 2 are green. When the roulette wheel is spun, the ball is equally likely to land on any of the 38 numbers. Suppose that you bet $1 on red. If the ball lands on a red number, you win $1; otherwise you lose your $1. Let X be the amount you win on your $1 bet. Then X is a random variable whose probability distribution is as follows: x 1 –1 P(X = x) 0.474 0.526 a) Verify that the probability distribution is correct. b) Find the expected value of the random variable X. c) On average, how much will you lose per play? -2- Math 160 Intro to Statistics OCC Section 5.2 – The Mean and Standard Deviation of a Discrete Random Variable Exercises: 4) Hosted by Howie Mandel, "Deal or No Deal" was a popular game show that debuted in 2000. The game revolves around the opening of a set of numbered briefcases, each of which contains a different prize (cash or otherwise). The contents (i.e., the values) of all of the cases are known at the start of the game, but the specific location of any prize is unknown. The contestant claims (or is assigned) a case to begin the game. The case's value is not revealed until the conclusion of the game. The following graphic to the right shows the 26 different denominations.The contestant then begins choosing cases that are to be removed from play. The amount inside each chosen case is immediately revealed; by process of elimination, the amount revealed cannot be inside the case the contestant initially claimed (or was assigned). Throughout the game, after a predetermined number of cases have been opened, the banker offers the contestant an amount of money and/or prizes to quit the game, the offer based roughly on the amounts remaining in play and the contestant's demeanor, the bank tries to 'buy' the contestant's case for a lower price than what's inside the case. The player then answers the titular question, choosing: "Deal", accepting the offer presented and ending the game, or "No Deal", rejecting the offer and continuing the game. This process of removing cases and receiving offers continues, until either the player accepts an offer to 'deal', or all offers have been rejected and the values of all unselected cases are revealed. Should a player end the game by taking a deal, a pseudo-game is continued from that point to see how much the player could have won by remaining in the game. Depending on subsequent choices and offers, it is determined whether or not the contestant made a "good deal", i.e. won more than if the game were allowed to continue. Since the range of possible values is known at the start of each game, how much the banker offers at any given point changes based on what values have been eliminated. To promote suspense and lengthen games, the banker's offer is usually less than the expected value dictated by probability theory, particularly early in the game. Generally, the offers early in the game are very low relative to the values still in play, but near the end of the game approach (or even exceed) the average of the remaining values. If you were to select a suitcase at random, what is the expected value? Expected Value = xP( X x) 1 1 1 1 1 3418416.01 ($.01) ($1) ($5) ($750000) ($1000000) $131, 477.54 26 26 26 26 26 26 -3-