If X (in millions) is u a random variable gamma (β = 2.5, α = 2)....
If X (in millions) is u a random variable gamma (β = 2.5, α = 2).
a. What is the probability of x being greater than 6.5 million?
b. If we take a sample of 10 and define T = ∑Xi. What is the probability that the total in this between 50 and 56.5 million?
c. If we take a sample of 30 and define W (average) = ∑Xi / 30. What is the probability that the average is between 4.5 and 5.5 million?
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If X (in millions) is u a random variable gamma (β = 2.5, α =
2)....
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
LetX-Gamma(α = 2, β = 4), Y-Gamma(α = 3, β = 4), X & Y are...
LetX-Gamma(α = 2, β = 4), Y-Gamma(α = 3, β = 4), X & Y are independent, Z,- , Z,-X + Y. 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 Z2, using the two variable transformation method. In addition, clearly write the support for this joint pdf. When done, your answer should include the expression c) (5 pts) You should see...
6.29. Let Xi, X2, , X,be a random sample from a gamma distribution with known parameter α-3 and unknown β > 0,' Discuss the construction of a confidence interval for B. Hint: what is the di...
6.29. Let Xi, X2, , X,be a random sample from a gamma distribution with known parameter α-3 and unknown β > 0,' Discuss the construction of a confidence interval for B. Hint: what is the distribution of 2 Σ x/P Follow the procedure outlined in Exercise 6.28.
6.29. Let Xi, X2, , X,be a random sample from a gamma distribution with known parameter α-3 and unknown β > 0,' Discuss the construction of a confidence interval for B. Hint: what...
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...
9. (5 marks) Consider a Gamma random variable, Y ~ Ganzma(α = n/2, β). Find the...
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?
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)...
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. (a) Suppose that x1,... , Vn are a random sample from a gamma distribution with...
2. (a) Suppose that x1,... , Vn are a random sample from a gamma distribution with shape parameter α and rate parameter λ, Here α > 0 and λ > 0. Let θ-(α, β). Determine the log-likelihood, 00), and a 2-dimensional sufficient statistic for the data (b) Suppose that xi, ,Xn are a random sample from a U(-9,0) distribution. f(x; 8) otherwise Here θ > 0, Determine the likelihood, L(0), and a one-dimensional sufficient statistic. Note that the likelihood should...
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 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...
2. (a) Suppose that xi,...,In are a random sample from a gamma distribution with shape parameter...
2. (a) Suppose that xi,...,In are a random sample from a gamma distribution with shape parameter and rate parameter λ, Γ(a) Here α > 0 and λ > 0. Let θ sufficient statistic for the data (α, β). Determine the log-likelihood, I(0), and a 2-dimensional b) Suppose that xi,...,In are a random sample from a U(-0,) distribution, 1/(20) if- otherwise x-θ f(x;0)-' 0, Here θ > 0, Determine the likelihood, L(0), and a one-dimensional sufficient statistic. Note that the likelihood...
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
LetX-Gamma(α = 2, β = 4), Y-Gamma(α = 3, β = 4), X & Y are independent, Z,- , Z,-X + Y. 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 Z2, using the two variable transformation method. In addition, clearly write the support for this joint pdf. When done, your answer should include the expression c) (5 pts) You should see...
6.29. Let Xi, X2, , X,be a random sample from a gamma distribution with known parameter α-3 and unknown β > 0,' Discuss the construction of a confidence interval for B. Hint: what is the distribution of 2 Σ x/P Follow the procedure outlined in Exercise 6.28.
6.29. Let Xi, X2, , X,be a random sample from a gamma distribution with known parameter α-3 and unknown β > 0,' Discuss the construction of a confidence interval for B. Hint: what...
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...
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?
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)...
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. (a) Suppose that x1,... , Vn are a random sample from a gamma distribution with shape parameter α and rate parameter λ, Here α > 0 and λ > 0. Let θ-(α, β). Determine the log-likelihood, 00), and a 2-dimensional sufficient statistic for the data (b) Suppose that xi, ,Xn are a random sample from a U(-9,0) distribution. f(x; 8) otherwise Here θ > 0, Determine the likelihood, L(0), and a one-dimensional sufficient statistic. Note that the likelihood should...
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...
2. (a) Suppose that xi,...,In are a random sample from a gamma distribution with shape parameter and rate parameter λ, Γ(a) Here α > 0 and λ > 0. Let θ sufficient statistic for the data (α, β). Determine the log-likelihood, I(0), and a 2-dimensional b) Suppose that xi,...,In are a random sample from a U(-0,) distribution, 1/(20) if- otherwise x-θ f(x;0)-' 0, Here θ > 0, Determine the likelihood, L(0), and a one-dimensional sufficient statistic. Note that the likelihood...
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