Let X, Y be independent random variables where X is binomial(n = 4, p = 1/3) and Y is binomial(n = 3,p = 1/3). Find the moment-generating functions of the three random variables X, Y and X + Y . (You may look up the first two. The third follows from the first two and the behavior of moment-generating functions.) Now use the moment-generating function of X + Y to find the distribution of X + Y .
Let X, Y be independent random variables where X is binomial(n = 4, p = 1/3)...
The moment generating function ф(t) of random variable X is defined for all values of t by et*p(x), if X is discrete e f (x)dx, if X is continus (a) Find the moment generating function of a Binomial random variable X with parameters n (the total number of trials) and p (the probability of success). (b) If X and Y are independent Binomial random variables with parameters (n1 p) and (n2, p), respectively, then what is the distribution of X...
Let X and Y be independent binomial random variables B(n,p) on the same sample space. Show that X + Y is also a binomial random variable B(?,?). What values should replace the questions marks?
Random variables X and Y have following distributions. P(X = -1) = 3/4, P(X = 3) = 1/4 P(Y = -3) = 1/2, P(Y = 2) = 1/2 a) Using the moment generating functions for random variables above find: E[X+Y) b) Using the moment generating functions for random variables above find: Var(X+Y)
Let Y1, Y2, . .. , Yn be independent and identically distributed random variables such that for 0 < p < 1, P(Yi = 1) = p and P(H = 0) = q = 1-p. (Such random variables are called Bernoulli random variables.) a Find the moment-generating function for the Bernoulli random variable Y b Find the moment-generating function for W = Yit Ye+ … + . c What is the distribution of W? 1.
Let X and Y be independent random variables uniformly distributed on the interval [1,2]. What is the moment generating function of X + 2Y? Let X and Y be independent random variables uniformly distributed on the interval [1,2]. What is the moment generating function of X + 2Y?
Let Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter p. Suppose that Y, X1 and X2 are independent. Proof using the de finition of distribution function that the the distribution function of Z =Y Xit(1-Y)X2 is F = pF14(1-p)F2 Don't use generatinq moment functions, characteristic functions) Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter...
Let X be a random variable which follows truncated binomial distribution with the following p.m.f. P(X=x) =((n|x)(p^x)(1−p)^(n−x))/(1−(1−p)^n), if x= 1,2,3,···,n. •Find the moment generating function (m.g.f.) and the probability generating function(p.g.f.). •From the m.g.f./p.g.f., and/ or otherwise, obtain the mean and variance. Show all the necessary steps for full credit.
Random variables X and Y have following distributions: PIX = -4) = 2/3, P(X = -1) = 1/3 PſY = 2) = 1/2, P(Y = 3) = 1/2 a) (5 points) Using the moment generating functions for the random variables above find: E[X+Y] b) (5 points) Using the moment generating functions for the random variables above find: Var(X+Y)
Random variables X and Y have following distributions. PIX = -1) = 3/4, P(X = -2) - 1/4 PLY = 3) = 1/2, PIY = 2) = 1/2 a) Using the moment generating functions for random variables above find: E[X+Y] b) Using the moment generating functions for random variables above find: Var(X+Y)
Show that if X follows a binomial distribution with n, trials and probability of success p,-p,jz 1,2, Hint: Use the moment generating function of Bernoulli random variable) 1. , n and X, are independent then X, follows a binomial distribution.