1. If the p.g.f. of a random variable X assuming non-negative values is Gx(s), then find...
If X is a nonnegative integer-valued random variable then the function P(z), defined for lzl s 1 by is called the probability generating function of X (a) Show that d* (b) With 0 being considered even, show that PX is even) = P(-1) + P(1) (c) If X is binomial with parameters n and p, show that Pix is even) -1+12p (d) If X is Poisson with mean A, show that 1+ e-24 2 P[X is even)- (e) If X...
Problem 1 (16 points). Suppose that X and Y are independent random variables and that Y follows a geometric distribution with parameter p. Assume that X takes only nonnegative integer values, and let Gx(z) be the probability generating function of X. (We make no additional assumptions about the distribution of X.) Show that P(X<Y) = Gx(1- p). Clearly indicate the step(s) in your argument that use the assumption that X and Y are independent.
Answer the following questions: (a) Suppose X is a uniform random variable with values 1, 2, 3, and 4. Then, 1) P(X = 3) = (correct to 2 decimal). 2) P(X S 3) = (correct to 2 decimal) 3) P(X > 3) = (correct to 2 decimal) 4) P(2 < X < 4) = (correct to 1 decimal) (b) Suppose Y is a random variable having Binomial distribution with parameters n = 10 and p = 0.5. Find (1) P(Y...
Let X be a zero-mean normal distributed random variable with variance of 2. Let Y gx), where 4 -2542-1 120 0, Find the CDF and PDF of the random variable Y.
Let X be a zero-mean normal distributed random variable with variance of 2. Let Y gx), where 4 -2542-1 120 0, Find the CDF and PDF of the random variable Y.
Question 1: 1a) Let the random variable X have a geometric distribution with parameter p , i.e., P(X = x) = pq??, x=1,2,... i) Show that P(X > m)=q" , where m is a positive integer. (5 points) ii) Show that P(X > m+n X > m) = P(X>n), where m and n are positive integers. (5 points) 1b) Suppose the random variable X takes non-negative integer values, i.e., X is a count random variable. Prove that (6 points) E(X)=...
P7
continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY...
Random variable X has MGF(moment generating function) gX(t) = , t < 1. Then for random variable Y = aX, some constant a > 0, what is the MGF for Y ? What is the mean and variance for Y ?
Problem 8 (10 points). Let X be the random variable with the geometric distribution with parameter 0 <p <1. (1) For any integer n > 0, find P(X >n). (2) Show that for any integers m > 0 and n > 0, P(X n + m X > m) = P(X>n) (This is called memoryless property since this conditional probability does not depend on m. Dobs inta T obabilita ndomlu abonn liaht bulb indofootin W
X is a negative binomial random variable with parameters. r=1 and P(S)=p p=62/100. Show that the probability mass function for x is well defined. That it satisfies the requirement for any discrete pmf