.Suppose X~Gamma(α,γ). What are the first and second population moments of X?
.Suppose X~Gamma(α,γ). What are the first and second population moments of 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
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~Gamma(α, β) Y|X ~ Poi(X) Compute E(Y) and VAR(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.
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...
Let X1, . . . , Xn be a sample taken from the Gamma distribution Γ(2, θ−1) with pdf f(x,θ)= θ^2xexp(−θx) if x ≥ 0, θ ∈ (0,∞), and 0 otherwise, (A) Show that Y = ∑ni=1 Xi is a complete and sufficient statistic. (B) Find E(1/Y) . Hint: If W ∼ χ2(k) then E(W^m) = 2mΓ(k/2+m) for m > −k/2. Note also that Y Γ(k/2) Γ(n) = (n − 1)!, n ∈ N∗ . Facts from 1(C) are useful:...
Let X ~ Gamma(k, β) and Y ~ Gamma(k, 1) Gamma( α, 3) Cx Show that Y = 스 is a pivot
Let X ~ Gamma(k, β) and Y ~ Gamma(k, 1) Gamma( α, 3) Cx Show that Y = 스 is a pivot
Suppose that x1, . . . , xn are a random sample from a B(α, β) distribution: f(x; α, β) = x^(α-1) (1-x)^(β-1) Here E[X] = α/(α + β) and E[X^2 ] = ((α + 1)α)/{(α + β + 1)(α + β)}. (a) Show that the method of moments, using the first two moments, gives the equations 0 = α(1 − m1 ) − βm1 m1 − m2 = α(m2 − m1 ) + βm2 (b) Determine the method of moments...
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?
00900 Gamma Distribution Exercise. To determine the variance of these estimators, compute the appropriate second derivatives. o? a2 θα β This give a Fisher information matrix /(α, β)::n( ) Inf(a) In r(a) - -a 1(0.19, 5.18) (0.19,5.18) 500 28.983 -0.193 500 NB. ψι(a) d2mT(a)/da2 is known as the trigamma function and is called in R with trigamma. 8/10 00090 Gamma Distribution The inverse matrix 1 0.0422 1.1494 /(α, β)--500 (1.1494 172.5587 Var(a) ~ 8.432 x 10-5 σ& 0.00918 Var(8) 0.3451...