Homework Answers
ANSWER:
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 show that W = 2Y/βhas a χ 2 distribution with n degrees of freedom.
1.Let 
follows a Gamma distribution with
parameters 
The available function is: 
-
The moment generating function of the
is,
Hence,
is 
moment generating function, Therefore,
this is the moment generating function 
for a chi-square 
variable with 
degrees of freedom.
is,
is 
moment generating function, Therefore,
this is the moment generating function 
for a chi-square 
variable with 
degrees of freedom.Add Answer to:
10. (9 pts) Suppose that Yhas a gamma distribution with a -n/2 for some positive integer...
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...
9. (9 pts) The random variable ??~??????????(∝= 2, ?? = 4). Use the method of moment-generating...
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...
A.6.6. We mentioned in class that the Gamma(, 2) distribution when k is a positive integer is cal...
Please answer A.6.6.:
The previous two questions mentioned above are included
below:
A.6.6. We mentioned in class that the Gamma(, 2) distribution when k is a positive integer is called the Chi-square distribution with k degrees of freedom. From the previous two problems, find the mean, variance, and MGF of the Chi-square distribution with k degrees of freedom. A.6.5. In class we showed that if X ~ Gamma(α, β) then E (X) = aß and uar(X) = αβ2 by using...
Find the moment generating function for the following distributions: N(μ, σ2), Poisson(λ), Gamma(α, β), Chi-square with...
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)
Having troubles with question 2. Please help 2. If X has a Gamma distribution with parameters...
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...
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. Let Yİ ~Gamma ( -3,ß-3), Y ~Gamma( -5, ß-1), and W-2% + 6K. a) (9...
3. Let Yİ ~Gamma ( -3,ß-3), Y ~Gamma( -5, ß-1), and W-2% + 6K. a) (9 pts) Find the moment generating function of W. Justify all steps b) (3 pts) Based on your result in part (a), what is the distribution of W(name and parameters)?
10) (11) Let X and Y be 2 independent random variables. Suppose X ~ Gamma(0, 38)...
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
9. Show that Y-Σ|-1 z? has a Chi-squared distribution with n degrees of freedom, where Z's...
9. Show that Y-Σ|-1 z? has a Chi-squared distribution with n degrees of freedom, where Z's are iid rvs from a standard normal distribution. (Prove it mathematically)
1. Suppose t hat Xhas t he chi-square distribution on p1∈(0, ∞) degrees of f reedom...
1. Suppose t hat Xhas t he chi-square distribution on p1∈(0, ∞) degrees of f reedom and that, i ndependently, Y has t he chi-square distribution on p2∈(0, p1) degrees of f ree-dom. a. Use moment generating functions to find the distribution of X + Y . b. A naive guess might be that the distribution of X − Y is chi-square on p1− p2 degrees of freedom. Prove that such a guess is wrong by demonstrating that P (X...
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...
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...
Please answer A.6.6.:
The previous two questions mentioned above are included
below:
A.6.6. We mentioned in class that the Gamma(, 2) distribution when k is a positive integer is called the Chi-square distribution with k degrees of freedom. From the previous two problems, find the mean, variance, and MGF of the Chi-square distribution with k degrees of freedom. A.6.5. In class we showed that if X ~ Gamma(α, β) then E (X) = aß and uar(X) = αβ2 by using...
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)
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...
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. Let Yİ ~Gamma ( -3,ß-3), Y ~Gamma( -5, ß-1), and W-2% + 6K. a) (9 pts) Find the moment generating function of W. Justify all steps b) (3 pts) Based on your result in part (a), what is the distribution of W(name and 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
9. Show that Y-Σ|-1 z? has a Chi-squared distribution with n degrees of freedom, where Z's are iid rvs from a standard normal distribution. (Prove it mathematically)
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