# 11 21 computing a pmf px x collect all possible

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Unformatted text preview: ust be TAIL) = P (TTTT ....H ) (tosses are independent) = (1 − p )k −1 p , M. Chen ([email protected]) ENGG2430C lecture 5 k = 1, 2, .... 11 / 21 Computing a PMF pX (x ) Collect all possible outcomes ω ∈ Ω, such that X (ω ) = x : {ω : ω ∈ Ω, X (ω ) = x } pX (x ) = P ({ω : ω ∈ Ω, X (ω ) = x }). Repeat for all x ∈ R . Example: two independent throws of a fair die. Deﬁne X and Y as the outcomes of the ﬁrst and second throw, respectively 9 Deﬁne Z = min(X ;Y ). pZ (2) = 36 M. Chen ([email protected]) ENGG2430C lecture 5 12 / 21 Binomial Random Variable Experiment: ﬂip a coin Independently for n times. Deﬁne Bernoulli/indicator r.v.s Xi , (i = 1, 2, . . . , n) be the outcome of the i -th ﬂip: 1, the ith toss gives HEAD; Xi = 0, the ith toss gives TAIL. The PMF is pXi (1) = p and pXi (0) = 1 − p . Let Y = ∑n=1 Xi . Then Y is a Binomial random variable. Its PMF is i given by n pY (k ) = P (Y = k ) = P ∑ Xi = k i =1 = (n ) p k (1 − p )n−k . k M. Chen ([email protected]) ENGG2430C lecture 5 13 / 21 Service Facility Design n: the number of customers p : probability that a customer requires service s : number of service lines X : a r.v. denoting the number of service requests Goal: Design a system that “no-service" probability is no more than ε : P (X > s ) = PX (s + 1) + · · · + pX (n) n n = ∑ k p k (1 − p )n−k . k =s +1 The larger the s , the smaller the “no-service" probability is. However, it costs more to run the service. Given ε , the optimal s ∗ (ε ) = min(s ∈ N : P (X > s ) ≤ ε ). M. Chen ([email protected]) ENGG2430C lecture 5 14 / 21 Expectation The so-called ﬁrst-order statistics about random outcomes: E [X ] = ∑ xpX (x ). (A weighted sum of outcomes x ’s.) x Interpretations: The center of gravity of the probability mass The average of repeated experiments Example: a sample space Ω = {0, 1, . . . , n} with a uniform probability law (From MIT open course 6.041 / 6.431 lecture notes.) E [X ] = 0 × M. Chen ([email protected]) 1 1 n n +1× +···+ = n+1 n+1 n+1 2 ENGG2430C lecture 5 15 / 21 Examples Calculate the expectation of a Bernoulli random...
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