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).
Find the moment generating function for the following distributions: N(μ, σ2), Poisson(λ), Gamma(α, β), Chi-square with...
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. (9 pts) The random variable r-Gamma(x-2, β-4). functions to prove that the moment generating function for the random variable W mw(t) (1-12t)2. Use the method of moment-generating 3Y +5is eSt 10, (9 pts) Suppose that Y has a gamma distribution with α-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- 2Y /g has a Chi-squared distribution with n degrees of freedom. Make sure you show...
(40) Draw the pdf and cdf of U(α, β) for any α, β. (41) Draw the pdf and cdf of Exp(A) for any λ. (42) Draw the pid and cdf of Garn(λ, α). Use α-9 and λ-1/2. (43) Draw the pdf and cdf of N(μ, σ2) for any μ and σ2. (44) Draw the pdf and cdf of N(0,1). (40) Draw the pdf and cdf of U(α, β) for any α, β. (41) Draw the pdf and cdf of Exp(A)...
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...
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?
Suppose we have 5 independent and identically distributed random variables Xi,X2.X3,X4,X5 each with the moment generating function 212 Let the random variable Y be defined as Y -XX. The density function of Y is (a) Poisson with λ-40 (b) Gamma with α-10 and λ-8 (c) Normal with μ-40 and σ-3.162 (d) Exponential with λ = 50 (e) Normal with μ-50 and σ2-15
4. The moment generating function of the normal distribution with parameters μ and σ2 is (t) exp ( μ1+ σ2t2 ) for -oo < t oo. Show that E X)-ψ(0)-μ and Var(X)-ψ"(0)-[ty(0)12-σ2. 5. Suppose that X1, X2, and X3 are independent random variables such that E[X]0 and ElX 1 for i-12,3. Find the value of E[LX? (2X1 X3)2] 6. Suppose that X and Y are random variables such that Var(X)-Var(Y)-2 and Cov(X, Y)- 1. Find the value of Var(3X -...
Find the joint pdf of the random sample Y1,...,Yn from the following distributions: a. Poisson(λ). b.N(0,σ2).
. Let Yi, ,Ý, be a sample from N(μ, σ2) distribution, where both μ and σ2 are un known Repeat the argument that was given in class to show that is a pivot (start by representing Yj as a linear function of a N(0, 1) random variable). Use the fact that (n-pe, of freedom") to construct the confidence interval with coverage probability 95% for σ2 (you can state the answer in terms of quantiles of X2-distribution, or find their numerical...
Given the common probability distributions & moment generating functions NOTES Is very desirable to be used in applications but both the PDF MGF Normal ALT/ Notation N(μ,σ) population mean μ and population st dev σ sample space Ω is defined or all X must be known. s an approximation to the normal dist for smaller samples, with degrees freedom v T dist N/A ALT/ Notation T PLEASE LET ME KNOW IF YOU FIND AN MGF Sample space is defined Forx>0...