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MATH 477 { Section E1 7/8/98 Summary of Random Variables Denition. A random variable is a real-valued function on a sample space. Denition. The cumulative distribution function (or c.d.f) of the random variable X is dened for all real numbers b, ,1 < b < 1, by F (b) = IP fX bg : A cumulative distribution function F has the following properties: 1. F is a nondecreasing function; that is, if a < b, then F (a) F (b). 2. blim F (b) = 1. !1 3. b!,1 lim F (b) = 0. 4. F is right continuous. That is, for any b and any decreasing seqence bn , n 1, that converges to b, limn!1 F (bn ) = F (b). Let F be the c.d.f. of random variable X , then IP fX bg = F (b) IP fa < X bg = F (b) , F (a) 1 IP fX < bg = nlim !1 F b , n Denition. Let X be a discrete random variable. Then, the probability mass function, p(a), is dened to be p(a) = IP fX = ag : The probability mass function p(a) is positive for at most a countable number of values of a. That is, if X must assume one of the values x1 ; x2; : : :, then p (xi ) > 0 i = 1; 2; : : : ; p(x) = 0 all other values of x. 1 Summary of Random Variables 7/8/98 The cumulative distribution function F can be expressed in terms of p(a) by F (a) = X a allx p(x) : Denition. If X is a discrete random variable having a probability mass function p(x), the expectation or the expected value of X , denoted by IE [X ], is dened by X IE [X ] = x:p(x)>0 x p(x) : Proposition 5.1. If X is a discrete random variable that takes on one of the values xi, i 1, with respective probabilities p (xi ), then for any real-valued function g X IE [g (X )] = g (xi ) p (xi ) : i Corollary 5.1. If a and b are constants, then IE [aX + b] = aIE [X ] + b : IE [X n ] = X x:p(x)>0 xn p(x). Denition. If X is a random variable with mean , then the variance of X , denoted by Var(X ), is dened by h i Var(X ) = IE (X , )2 : h i Var(X ) = IE X 2 , (IE [X ])2. Var(aX + b) = a Var(X ) . 2 Denition. The square root of the Var(X ) is called the standard deviation of X and we denote it by SD(X ). That is, q SD(X ) = Var(X ) : 2 Summary of Random Variables 7/8/98 Bernoulli Distribution. If X is distributed as a Bernoulli random variable with parameter p, then p(0) = IP fX = 0g = 1 , p p(1) = IP fX = 1g = p and Var(X ) = p(1 , p) : IE [X ] = p Binomial Distribution. If X is distributed as a binomial random variable with parameters n and p, then p(i) = and ! n i p (1 , p)n,i i Var(X ) = np(1 , p) : IE [X ] = np Probability mass function for a binomial random variable with parameters n = 10 and p = 0:1: 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 4 6 8 10 Probability mass function for a binomial random variable with parameters n = 10 and p = 0:5: 0.5 0.4 0.3 0.2 0.1 0 2 4 6 3 8 10 Summary of Random Variables 7/8/98 Proposition 7.1. If X is a binomial random variable with parameters (n; p), where 0 < p < 1, then as k goes from 0 to n, IP fX = kg rst increases monotonically and then decreases monotonically, reaching its largest value when k is b(n + 1)pc. Geometric Distribution. If X is distributed as a geometric random variable with parameter p, then p(n) = (1 , p)n,1 p and Var(X ) = 1 p,2 p : 1 IE [X ] = p Probability mass function for a geometric random variable with parameter p = 1=9: 0.2 0.15 0.1 0.05 0 10 20 30 40 50 Negative Binomial Distribution. If X is distributed as a negative binomial random variable with parameters r and p, then p(n) = and IE [X ] = r p ! n,1 r p (1 , p)n,r r,1 Var(X ) = r(1p,2 p) : 4 7/8/98 Summary of Random Variables Probability mass function for a negative binomial random variable with parameters p = 1=9 and r = 2: 0.08 0.06 0.04 0.02 0 10 20 30 40 50 Poisson Distribution. If X is distributed as a Poisson random variable with parameter , then p(i) = and IE [X ] = e, i i! Var(X ) = : A Poisson random variable approximates a binomial random variable with = large. Probability mass function for a Poisson random variable with parameter = 0:1: 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 2 3 5 4 5 np and n Summary of Random Variables Probability mass function for a Poisson random variable with parameter = 1: 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 Probability mass function for a Poisson random variable with parameter = 5: 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 5 10 15 20 Probability mass function for a Poisson random variable with parameter = 10: 0.25 0.2 0.15 0.1 0.05 0 5 10 6 15 20 7/8/98