9. (9 pts) The random variable r-Gamma(x-2, β-4). functions to prove that the moment generating function...
9. (9 pts) The random variable ??~??????????(∝= 2, ?? = 4). Use the method of moment-generating functions to prove that the moment generating function for the random variable ?? = 3?? + 5 is 10. 9. (9 pts) The random variable Y-Gamma(α-2. functions to prove that the moment generating function for the random variable W mw(t)120)2 4). Use the method of moment-generating 3Y 5 is est (1-12t)2 10, (9 pts) Suppose that Y has a gamma distribution with α-n/2 for...
10. (9 pts) Suppose that Yhas a gamma distribution with a -n/2 for some positive integer n and β equal to some specified value. Use the method of moment-generating functions to prove that W 2% has a Chi-squared distribution with n degrees of freedom. Make sure you show all steps and give reasons for each one.
Find the moment generating function for the following distributions: N(μ, σ2), Poisson(λ), Gamma(α, β), Chi-square with k degrees of freedom, and Geometric(p). Question 7: Find the moment generating function for the following distributions: N(Lơ2 Poisson(A), Gamma(α, β), Chi-square with k degrees of freedom, and Geometric(p)
Let Y_1~Gamma(α=3,β=3), Y_2~Gamma(α=5,β=1), and W=2Y_1+6Y_2. a) (9 pts) Find the moment generating function ofW Justify all steps b) (3 pts) Based on your result in part (a), what is the distribution of W(name and parameters)? n 2N(O, I) 2. IfZ NO, 1), then Ux(1) 3. ItY Gmmaa,B) and W then Wx(n) - s, and i-1 7. y's~ Poisson(W (i-l, ,Rind) and U-ŽYi, then U-Poisson(XA) 8 If%-Gamma(a, β) (i-I, ,Rind) and U-ΣΥί , then U~Gamma( ,4 β).(Note: all same β) 9...
9. (5 marks) Consider a Gamma random variable, Y ~ Ganzma(α = n/2, β). Find the moment- generating function of U = c Y. If U ~ , what is c?
calculate the moment generating function for a random variable which has exponential distribution with parameter gamma.
O A Gamma random variable x with parameters xson Bo has the following moment generating function: M(t) = (1 - 45 for taß (9 pts) . Use the reg.f. Mit given above E(X4), (where X is a Canan c.v.) to calculate (9 pts) [ Use the mg.f. Mit given above to calculate the standard deviation of a Gamma rov. X. (9 pts) © The skewness of a r.V. X, with mear=u& std. dev. = T, is given by : skewness...
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...
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
Prove that if random variable X follows a standard normal distribution (with mean u= 0 and standard deviation o = 1), then Y = X2 follows a chi-square distribution with 1 degree of freedom. In particular, show that My(t) = Mx2(t) = E[etX?), which equals the moment generating function of a chi-square distribution with 1 degree of freedom.