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.
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9. (9 pts) The random variable ??~??????????(∝= 2, ?? = 4). Use
the method of moment-generating...
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...
10. (9 pts) Suppose that Yhas a gamma distribution with a -n/2 for some positive integer...
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...
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)
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?
6. (11 pts) Use the Distribution Function Method here: The random variable ??~????????(∝= 4, ?? =...
6. (11 pts) Use the Distribution Function Method here: The
random variable ??~????????(∝= 4, ?? = 2). Let ?? = ??4. Find the
pdf of U.
(1 l pts) Use the Distribution Function Method here: The random variable Y~Beta(α= 4, β = 2). Let U 6. Y4, Find the pdf of U.
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...
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc...
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X...
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...
Prove that if random variable X follows a standard normal distribution (with mean u= 0 and...
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.
calculate the moment generating function for a random variable which has exponential distribution with parameter gamma.
calculate the moment generating function for a random variable which has exponential distribution with parameter gamma.
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...
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)
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?
6. (11 pts) Use the Distribution Function Method here: The
random variable ??~????????(∝= 4, ?? = 2). Let ?? = ??4. Find the
pdf of U.
(1 l pts) Use the Distribution Function Method here: The random variable Y~Beta(α= 4, β = 2). Let U 6. Y4, Find the pdf of U.
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...
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X...
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...
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.
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