Homework Answers
2. LetX-Gamma(α = 2, β = 4), Y-Gam ma (α = 3, β = 4), X...
2. LetX-Gamma(α = 2, β = 4), Y-Gam ma (α = 3, β = 4), X & Y are independent, Z1 = , Z,-X + Y a) (3 pts) State the joint pdf ofX and Y. Simplify the expression, clearing all Г's. b) (9 pts) Find the joint pdf of Zi and Zz, using the two variable transformation method. In addition, clearly write the support for this joint pdf. When done, your answer should include the expression (5 pts) You...
2. LetX~Gamma(α = 2, β = 4), Y~Gamma(α = 3, β = 4), X & Y...
2. LetX~Gamma(α = 2, β = 4), Y~Gamma(α = 3, β = 4), X & Y are independent, Z,-x+r, Z,-X + Y a) (3 pts) State the joint pdf oEX and Y. Simplify the expression, clearing all b) (9pts) Find the joint pdf of Z and Z, using the two variable transformation method. In addition, clearly write the support for this joint pdf. When done, your answer should include the expression Z1Z21,2)2048 2048 11 )24e-22/4 c) (5 pts) You should...
Problem 1. (5 marks. 3. 2) Assume X ~ Gamma(01, β) and Y ~ Gamma(O2, β)...
Problem 1. (5 marks. 3. 2) Assume X ~ Gamma(01, β) and Y ~ Gamma(O2, β) are independent random variables. a) Compute the Joint density of U = X + Y and V X X + Y , be sure to include the support/domain. b) Based on the joint density derived in part (a) find the marginal densities of U and V, be sure to include the support (s)/domain(s). Explicitly state the name of the distributions of U and V...
Suppose X and Y are independent and Prove the following a) U=X+Y~gamma(α + β,γ) b) V=X/(X...
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
Let X ~ Gamma(k, β) and Y ~ Gamma(k, 1) Gamma( α, 3) Cx Show that Y = 스 is a pivot
Let X ~ Gamma(k, β) and Y ~ Gamma(k, 1) Gamma( α, 3) Cx Show that Y = 스 is a pivot
Let X ~ Gamma(k, β) and Y ~ Gamma(k, 1) Gamma( α, 3) Cx Show that Y = 스 is a pivot
Suppose that X~Gamma(α, β) Y|X ~ Poi(X)
Suppose that X~Gamma(α, β) Y|X ~ Poi(X) Compute E(Y) and VAR(Y)
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a) Suppose that α 4 is known and β is unknown. Find a complete sufficient statistic for β. Find the MVUE of β....
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a) Suppose that α 4 is known and β is unknown. Find a complete sufficient statistic for β. Find the MVUE of β. (Hint: What is E(Y)?) (b) Suppose that β 4 is known and α is unknown. Find a complete sufficient statistic for a.
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a)...
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...
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...
3. Suppose X ~ Beta(a, β) with the constants α, β > 0, Define Y- 1-X....
3. Suppose X ~ Beta(a, β) with the constants α, β > 0, Define Y- 1-X. Find the pdf of Y
3. A random variable X is said to have a Cauchy(α, β) distribution if and only...
3. A random variable X is said to have a Cauchy(α, β) distribution if and only if it has PDF function Now, suppose that Xi and X2 are independent Cauchy(0, 1) random variables, and let Y = X1 + X2. Use the transformation technique to find and identify the distribution of Y by first finding the joint distribution of Xi and Y. (Seahin 3 4
2. LetX-Gamma(α = 2, β = 4), Y-Gam ma (α = 3, β = 4), X & Y are independent, Z1 = , Z,-X + Y a) (3 pts) State the joint pdf ofX and Y. Simplify the expression, clearing all Г's. b) (9 pts) Find the joint pdf of Zi and Zz, using the two variable transformation method. In addition, clearly write the support for this joint pdf. When done, your answer should include the expression (5 pts) You...
2. LetX~Gamma(α = 2, β = 4), Y~Gamma(α = 3, β = 4), X & Y are independent, Z,-x+r, Z,-X + Y a) (3 pts) State the joint pdf oEX and Y. Simplify the expression, clearing all b) (9pts) Find the joint pdf of Z and Z, using the two variable transformation method. In addition, clearly write the support for this joint pdf. When done, your answer should include the expression Z1Z21,2)2048 2048 11 )24e-22/4 c) (5 pts) You should...
Problem 1. (5 marks. 3. 2) Assume X ~ Gamma(01, β) and Y ~ Gamma(O2, β) are independent random variables. a) Compute the Joint density of U = X + Y and V X X + Y , be sure to include the support/domain. b) Based on the joint density derived in part (a) find the marginal densities of U and V, be sure to include the support (s)/domain(s). Explicitly state the name of the distributions of U and V...
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
Let X ~ Gamma(k, β) and Y ~ Gamma(k, 1) Gamma( α, 3) Cx Show that Y = 스 is a pivot
Let X ~ Gamma(k, β) and Y ~ Gamma(k, 1) Gamma( α, 3) Cx Show that Y = 스 is a pivot
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a) Suppose that α 4 is known and β is unknown. Find a complete sufficient statistic for β. Find the MVUE of β. (Hint: What is E(Y)?) (b) Suppose that β 4 is known and α is unknown. Find a complete sufficient statistic for a.
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a)...
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...
3. Suppose X ~ Beta(a, β) with the constants α, β > 0, Define Y- 1-X. Find the pdf of Y
3. A random variable X is said to have a Cauchy(α, β) distribution if and only if it has PDF function Now, suppose that Xi and X2 are independent Cauchy(0, 1) random variables, and let Y = X1 + X2. Use the transformation technique to find and identify the distribution of Y by first finding the joint distribution of Xi and Y. (Seahin 3 4
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