Let X, X2, ..., X, be independent with X-Gamma (a,b). Let Y = EX. Prove that...
Suppose X and Y are independent and
Prove the following
a) U=X+Y~gamma(α + β,γ)
b) V=X/(X + Y ) ∼ beta(α,β)
c) U, V independent
d) ~gamma(1/2,
1/2) when W~N(0,1)
X ~ gammala, y) and Y ~ gamma(6, 7) We were unable to transcribe this image
5-4. Prove that the MGF of Gamma distribution is В f(t) В — t. 5-5. Let X1, X2,... ,X be independent with X~ Gamma (ai,ß). Let Y = EX4. Prove that Y Gamma (Eai,ß)
Let X1 and X2 be independent gamma distribution random variables with gamma (a1,1) and gamma (a2, 1). Find the marginal distributions of x1/(x1+x2) and x2/(x1+x2).
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?
Problem 5) Let X and Y be independent gamma RVs with parameter (a, 1) and (3, 1), respec- tively. a) Show that X + Y is also gamma RV with parameters (a +3,1). b) Compute the joint density of U = X + Y and V = ty
5. Let X; (i = 1, 2, 3) be be independent gamma random variables with a; = i and B. = 8. a. Find a maximum likelihood estimator of 8 and prove that it is unbiased. b. Show that 2(X1+X2+Xa) is a pivotal quantity for 0. c. Find a 95% confidence interval for 6.
Let X and Y be independent identically distributed random variables with means µx and µy respectively. Prove the following. a. E [aX + bY] = aµx + bµy for any constants a and b. b. Var[X2] = E[X2] − E[X]2 c. Var [aX] = a2Var [X] for any constant a. d. Assume for this part only that X and Y are not independent. Then Var [X + Y] = Var[X] + Var[Y] + 2(E [XY] − E [X] E[Y]). e....
10) (11) Let X and Y be 2 independent random variables. Suppose X ~ Gamma(0, 38) and Y ~ Gamma(a, 2B). Let 2 = 2X +3Y. Determine the probability distribution of Z. (Hint: use the method of moment-generating functions
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?
Below are sample questions: [5] 6. Let X ~ GAMMA(0,k). Prove that Y = 2x 2K)