Suppose X ∼ Gamma(a, b) and Y ∼ Gamma(c, d). Let W = X
+ Y .
(a) Find the MGF of W.
(b) What restrictions would need to be placed on the values of a,
b, c, and d in order for W
to be a Gamma Random Variable. What would the parameters be?
1. Suppose X ∼ Gamma(a,b) and Y ∼ Gamma(c,d). Furthermore suppose X and Y are independent. Let W = X + Y . (a) Find the MGF of W. (b) What restrictions would need to be placed on the values of a, b, c, and d in order for W to be a Gamma Random Variable. What would the parameters be?
Suppose X Gamma (a; b) and YGamma (c; d). Let W-X+Y. (a) Find the MGF of w. (b) What restrictions would need to be placed on the values of a, b; c; and d for Ww to be a Gamma Random Variable. What would the parameters be?
uppose XGamma(a, b) and Y Gamma(c,d). Let W -X +Y. (a) Find the MGF of W. b) What restrictions would need to be placed on the values of a, b, c, and d in order for W to be a Gamma Random Variable. What would the parameters be?
In details plz Thank you! Suppose that X ~ Gamma(a, b) and Y ~ Chisquare(k) and X and Y are independent. Îet w = X+Y (a) Find the MGF of W. / (b) For what value(s) of b would W be a Gamma Random Variable? What would its parameters be? (c) For what value(s) of b and a would W be a ChiSquare Random Variable? What would its parameter be? (d) For what value(a) of b, a, and k would...
Having troubles with question 2. Please help 2. If X has a Gamma distribution with parameters a and B, then its mgf is given by (a) Obtain expressions for the moment-genérating functions of an exponential random variable and of a chi-square random variable by recognizing that these are special cases of a Gamma distribution and using the mgf given above. (b) Suppose that X1 is a Gamma variable with parameters α1 and β, X2 is a Gamma variable with parameters...
7. The Gamma distribution is commonly used to model continuous data. The probability density function of a Gamma random variable is f (zlo, β)- a. Find the MGF of a Gamma random variable. b. Use the MGF to find the mean of a Gamma random variable. c. Use the MGF to find the second raw moment of a Gamma random variable. d. Use results (b) and (c) to find the variance of a Gamma random variable. e. Let Xi, í...
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X...
(1 point) In Unit 3, I claimed that the sum of independent, identically distributed exponential random variables is a gamma random variable. Now that we know about moment generating functions, we can prove it. Let X be exponential with mean A 4. The density is 4 a) Find the moment generating function of X, and evaluate at t 3.9 The mgf of a gamma is more tedious to find, so l'll give it to you here. Let W Gamma(n, A...
1. Suppose that Y ∼ Gamma(α, β) and c > 0 is a constant. (a) Derive the density function of U = cY. (b) Identify the distribution of U as a standard distribution. Be sure to identify any parameter values. (c) Can you find the distribution of U using MGF method also? I. Suppose that Y ~ Gamma(α, β) and c > 0 is a constant. (a) Derive the density function of U cY. (b) Identify the distribution of U...
STAT 140 Suppose that X have a gamma distribution with parameters a = 2 and θ= 3, and suppose that the conditional distribution of Y given X=x, is uniform between 0 and x. (1) Find E(Y) and Var(Y). (2) Find the Moment Generating Function (MGF) of Y. What is the distribution of Y?