Suppose that X has a gamma distribution with parameters α > 0 and
β>0. Show that if a is any value so that α+a>0 then E[X^a] = (β^aΓ(α + a))/Γ(a)
Suppose that X has a gamma distribution with parameters α > 0 and β>0. Show that...
3. Suppose that X has the gamma distribution with parameters α and β. (a) Determine the mode of X. (Be careful about the range of a) (b) Let c be a positive constant. Show that cX has the gamma distribution with parar neters and ß/c.
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...
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)...
Y1, Y2, ... Yn are a random sample from the Gamma distribution with parameters α and β (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 a is unknown. Find a complete sufficient statistic for α.
Two questions exist : ) if it has pdf. A railon variable X has the l'areio distril illi ribution with parameters m, a (m, α > 0 w 0 otherwise Show that if X has this Pareto distribution, then the random variable log(X/m) has the expo- nential distribution with parameter α Let X ~ Gamma(α, β), where α > 1 . Find E[1/X]. ) if it has pdf. A railon variable X has the l'areio distril illi ribution with parameters...
Let X be a R.V. with a gamma distribution and the following parameters (X~(α, 1)). What is the pdf and the cdf of Y = X/β, where β > 0 . What is the name of this type of distribution?
Let X1 ,……, Xn be a random sample from a Gamma(α,β) distribution, α> 0; β> 0. Show that T = (∑n i=1 Xi, ∏ n i=1 Xi) is complete and sufficient for (α, β).
Exercise: Let Yİ,Y2, ,, be a random sample from a Gamma distribution with parameters and β. Assume α > 0 is known. a. Find the Maximum Likelihood Estimator for β. b. Show that the MLE is consistent for β. c. Find a sufficient statistic for β. d. Find a minimum variance unbiased estimator of β. e. Find a uniformly most powerful test for HO : β-2 vs. HA : β > 2. (Assume P(Type!Error)- 0.05, n 10 and a -...
1. Suppose that Y ∼ Gamma(α, β) and c > 0 is a constant. (a) Derive the density function of U = cY. (b) Identify the distribution of U as a standard distribution. Be sure to identify any parameter values. (c) Can you find the distribution of U using MGF method also? I. Suppose that Y ~ Gamma(α, β) and c > 0 is a constant. (a) Derive the density function of U cY. (b) Identify the distribution of U...
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