6.2 DISCRETE PROBABILITY DISTRIBUTIONS Defn If a sample space contains a finite number of possibilities or an unending sequence with as many elements as there are whole numbers, it is called a discrete sample space. Defn A random variable defined over a discrete sample space is called a discrete random variable.Defn A table or formula listing all possible values that a discrete random variable can take on, along with the associated probabilities, is called a discrete probability distribution. Remark The probabilities associated with all possible values of a discrete random variable must sum to 1. Examples: 1. For Experiment No. 1, the discrete probability distributions of the random variables Xand Yare X 0 1 2 3 P(X=x) 1/8 3/8 3/8 1/8 Y -3 -1 1 3 P(Y=y) 1/8 3/8 3/8 1/8 2. Construct the discrete probability distribution for the random variable Mdefined in Experiment No. 2.

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6.3 EXPECTED VALUES DefnLet Xbe a discrete random variable with probability distribution x x1x2. . . xnP(X=x) f(x1) f(x2) . . . f(xn) The mean or expected value of X is niiiXxfxXE1)()(Examples: 1.Find the mean of the random variables Xand Yof Experiment No. 1. X 0 1 2 3 P(X=x) 1/8 3/8 3/8 1/8 )(XEX(0)(1/8) + (1)(3/8) + (2)(3/8) + (3)(1/8) = 12/8 or 1.5 Y -3 -1 1 3 P(Y=y) 1/8 3/8 3/8 1/8 )(YEY(-3)(1/8) + (-1)(3/8) + (1)(3/8) + (3)(1/8) = 0 2. Find the expected number of correct matches in Experiment No. 2. 3. In a gambling game a man is paid P50 if he gets all heads or all tails when 3 coins are tossed, and he pays out P30 if either 1 or 2 heads show. What is his expected gain?

ThmLet Xbe a discrete random variable with probability distribution x x1x2. . . xnP(X=x) f(x1) f(x2) . . . f(xn) The mean or expected value of the random variable g(X) is niiixfxgXgE1)()()]([Defn Let Xbe a random variable with mean μthen the variance of Xis 22)()(XEXVXDefn Let Xbe a discrete random variable with probability distribution x x1x2. . . xnP(X=x) f(x1) f(x2) . . . f(xn) The variance of Xis niiiXxfxXEXV1222)()()()(Thm Computational Formula for 2X2XVar(X) = E(X2) - [E(X)]2Example In Experiment No. 1, find the variance of X. Using the definition of Var(X), XE(X) = 1.5 2X(0-1.5)2(1/8) + (1-1.5)2(3/8) + (2-1.5)2(3/8) + (3-1.5)2(1/8) = 0.75Using the computational formula of the Var(X), E(X2) = 3 2X3 –(1.5)2= 0.75

Binomial Distribution Defn A binomial experiment is one that possesses the following properties: the experiment consists of nidentical trials each trial results in one of two outcomes, a “success” or a “failure”the probability of success on a single trial is equal to pand remains the same from trial to trial. The probability of a failure is equal to q=1-p. the trials are independent The random variable of interest X, the number of successes observed in ntrials, is called a binomial random variable.