8. The Gamma(a, A) distribution has density f(x)(a) where for a0' a 0 and A >...
Suppose that X1,..., Xn is a random sample from a gamma distribu- tion, The gamma distribution has parameters r and λ, and also has E(X)-r/λ and Var(X)-r/ P. Calculate the method of moments MOM) estimators of r and λ in terms of the first two sample moments Mi and M2 Suppose that X1,..., Xn is a random sample from a gamma distribu- tion, The gamma distribution has parameters r and λ, and also has E(X)-r/λ and Var(X)-r/ P. Calculate the...
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)
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume the prior distribution of Of λ is is an exponential with mean 1, i.e. the prior pdi g(A) e-λ, λ > 0 Note that the exponential distribution is a special gamma distribution; and a general gamma distribution with parameters α > 0 and β > 0 has the pd.f. h(A; α, β)-16(. otherwise Also the mean of a gamma random variable with the pd.f.h(Χα,...
5. The Exponential(A) distribution has density f(x) = for x<0' where λ > 0 (a) Show/of(x) dr-1. (b) Find F(x). Of course there is a separate answer for x 2 0 and x <0 (c Let X have an exponential density with parameter λ > 0 Prove the 'Inemoryless" property: P(X > t + s|X > s) = P(X > t) for t > 0 and s > 0. For example, the probability that the conversation lasts at least t...
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...
12.5A e 2 Suppose that A has a Gamma distribution: fA(A) 「3.5)23.5 (a) Suppose that the conditional distribution of X given Λ = λ is fxA(TA z ) e- for x > 0. i. Find Ex ii. Find Var( (b) Suppose that the conditional distribution of X given A = λ is frA (zA)-Xe-k for x > 0. Find the unconditional probability density function fx(x) of *
Recall that the exponential distribution with parameter A > 0 has density g (x) Ae, (x > 0). We write X Exp (A) when a random variable X has this distribution. The Gamma distribution with positive parameters a (shape), B (rate) has density h (x) ox r e , (r > 0). and has expectation.We write X~ Gamma (a, B) when a random variable X has this distribution Suppose we have independent and identically distributed random variables X1,..., Xn, that...
STAT 140 Suppose that X have a gamma distribution with parameters a = 2 and θ= 3, and suppose that the conditional distribution of Y given X=x, is uniform between 0 and x. (1) Find E(Y) and Var(Y). (2) Find the Moment Generating Function (MGF) of Y. What is the distribution of Y?
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.
Exercise 8 The pdf of Gamma(α, λ) is f(x)-ra)r"-le-Az for x 0. a. Let X ~ Gamma (a, λ). Show that E( )--A for α > 1 b. Let Ux2. Show that E()for n > 2 n-2