MATH 396 Lecture Notes - Lecture 1: Independent And Identically Distributed Random Variables, Bernoulli Trial, Probability-Generating Function
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3 main generating functions useful in deriving properties of random variables (r. vs). (i) Characteristic functions (ii) moment generating functions (m. g. f) (iii) ), and expectations (ii) definition: e(x) = sum. Non-negative integer-valued discrete r. vs. (i) the pmf is px(k) = p(x = k), for k x {0,1,2,3,4,}. Basic properties: pk (cid:951) 0 (or > 0 for k x) and p. Expectations always add: e(g1(x) + g2(x)) = e(g1(x)) + e(g2(x)) var(x) = e(x2) (e(x))2 = Four standard integer-valued non-negative discrete random variables: over k g(k)pk, provided the sum over. Result: e(g(x)) = sum k |g(k)| pk is finite. over k kpk. k pk = 1 (i) binomial: the number of successes in n independent trials, when each trial has probability of success p. Arises as the limit of a bin(n,p) and n , p 0, s. t. np . (ii) since, pk (cid:951) 0 and p. 0 (t) = e(xtx 1); 0(0) = p1; x (i) definition, for r. v.
