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Summary of Chapter Three Random Variables: Let S be the sample space of an experiment. A real-valued function X : S → R is called a random variable.If X takes finite or countable number of possible values, then X is said to be discrete. Distribution Function: For a random variable X, its distribution function F (x) is defined by F (x) = P (X ≤ x). Probability Mass Functions, Expectation and Variance: Let X be a discrete random variable with a space S = {x1, x2, · · · } and a probability mass function f (x) = P (X = x), then (1) X f (x) = 1 x∈S (2) F (x) = X f (x) y≤x,y∈S (3) µ = E(X) = X xf (x), x∈S 2 E(X ) = X x2f (x) x∈S (4) σ 2 = V (X) = E[(X − µ)2] = E(X 2) − µ2, p σ = V (X)(standard deviation). 1 (5) E[h(X)] = X h(x)f (x). x∈S Special Discrete Random Variables: (i)Let X be the number of successes in n Bernoulli trials (with probability p for a success in each trial). Then, X is a binomial random variable with parameters (n, p). Its probability mass function is given by n x f (x) = P (X = x) = p (1 − p)n−x , x = 0, 1, · · · , n x Its mean (or expected value) and variance are given by E(X) = np, V (X) = np(1 − p) (ii)Let X be the number of trials needed to observe the first success. Then X is a geometric random variable with parameter p. Its probability mass function is given by f (x) = p(1 − p)x−1, x = 1, 2, · · · Its mean and variance are given by 1 1−p E(X) = , V (X) = 2 . p p (iii) Let X be a hypergeometric random variable with parameters n, N, K. Its probability mass function is given by K N −K f (x) = x n−x N n , x = max{0, n + K − N }to min{K, n}, 2 With p = K/N , its mean and variance are given by E(X) = np, V (X) = N −n np(1 − p) N −1 (v) Let X be a Poisson random variable with parameter λ. Its probability mass function is given by e−λλx f (x) = , x = 0, 1, 2, · · · x! Its mean (or expected value) and variance are given by E(X) = V (X) = λ. 3