Suppose XPoisson(5) and Y Poisson(10), and they are independent. Using the moment generating function method, find...
5. Suppose X ~ Poisson(A = 5) and Y ~ Poisson(λ = 10), and they are independent. Using the moment generating function method, find the distribution of Z = X + Y.
2. Consider the Poisson distribution, which has a pdf defined as: a) Derive the moment generating function. b) Use the moment generating function and the method of moments to find the mean and the variance. c) If X follows the Poisson distribution with Xx - 2.3, and Y follows a Poisson distribution with XY-54, what is the distribution of the sum X + Y, assuming that X and Y are independent?
5 (10 points) X and Y are independent random variables with common moment generating function M(t) eT. Let W X + Y and Z X - Y. Determine the joint moment generating function, M(ti, t2) of W and Z Find the moment generating function of W and Z, respectively
Let X, Y and Z be three independent Poisson random variable with parameters λι, λ2, and λ3, respectively. For y 0,1,2,t, calculate P(Y yX+Y+Z-t) (Hint: Determine first the probability distribution of T -X +Y + Z using the moment generating function method. Moment generating function for Poisson random variable is given in earlier lecture notes) Let X, Y and Z be three independent Poisson random variable with parameters λι, λ2, and λ3, respectively. For y 0,1,2,t, calculate P(Y yX+Y+Z-t) (Hint:...
Problem 3 Let X be Uniform(0,1) and Y be Exponential (1). Assume that X and Y are independent. i. Find the PDF of Z- X +Y using convolution. ii. Find the moment generating function, øz(s), of Z. Assume that s< 0. iii. Check that the moment generating function of Z is the product of the moment gen erating functions of X and Y Problem 3 Let X be Uniform(0,1) and Y be Exponential (1). Assume that X and Y are...
I. Suppose that χ ~ Poisson (2) and y ~ Poisson (3) are independent random variables. (a) Find the probability generating function of χ + y. (b) Use part (a) to find P(χ + y = 13). 2. Suppose that χ ~ Poisson (2) and y ~ Geom(0.25) are independent random variables. (a) Find the probability generating function of . (b) Find the probability generating function of χ + y.
Suppose Y-X1-X2 where X1, x2 are iid Poisson(11) (a) Show that Y has moment generating function My (t) = e11(ette-t-2) (b) Even though you can do it from other results, use the mgf in (a) to find Var(Y).
(1 point) Suppose that the moment generating function of a random variable X is My(t) = exp(4e – 4) and that of a random variable Y is My(t) = ( oer + 3)''. If X and Y are independent, find each of the following. (a) P{X + Y = 2} = (b) P{XY = 0} = (c) E[XY] = (d) E[(X+Y)?] =
1. Using the appropriate moment generating,function. Show that Var(X)-: ? when Poisson distribution with mean ?. X has the ting function of the random variable with probability density function
Suppose that X 1 has a Poisson distribution with mean 2, X 2has a Poisson distribution with mean 3 , X 3 has a Poisson distribution with mean 5 and that X 1 , X 2 and X 3 are independent. Define Y = X 1 + X 2 + X 3. Determine the moment-generating function for Y.