If X has a Standard Gamma distribution with α=7.
as for standard gamma distribution: =1
a)
E(X)= α =7*1 =7
b) σx= sqrt(α 2)=sqrt(7*12)=2.6458
c)
P(X<=5)=0.237817
d)
P(X>8)=0.313374
e)
P(3<X<8)=0.653117
If X has a Standard Gamma distribution with α=7. Compute E(X)=? Compute σx=? Compute P(X≤5)=? Compute...
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 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)
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8. The Gamma(a, A) distribution has density f(x)(a) where for a0' a 0 and A > 0 (a) Showfx,of(x) dr-1. Recall !"(a)-C"rtta-idt. (b) If X has a gamma distribution with parameters α and λ, find a general expression for E(Xk). (Answer: ) (c) Use your answer to the last question to find Var(X). The identity「(α + 1) a「(a) will help.
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...
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