※ 引述《ashlem (再見了，大頭)》之銘言：
: 1. Three fair dice are cast. In 10 independent casts, let
: X be the number of times all three faces are alike.
: Y be the number of times only two faces are alike.
: Find the joint p.d.f. of X and Y and compute E(6XY).
: (Ans: E(6XY) = 25/4)
: Note: Are X and Y independent
X~bin( 10, (1/6)^2 )
Y~bin( 10, 15*(1/6)^2 )
10 10-x 2x 2 y 10-x-y
P(X,Y)=P(X=x,Y=y)= C C (1/6) (15*(1/6) ) (1-16/6^2)
x y
~Multi(10,1/6^2,15/6^2)
E(6XY)=6E[XY]=6(Cov(X,Y)+E[X]E[Y]) where Cov(X,Y)=-n*p_x*p_y , E[X]=n*p_x ,
E[Y]=n*p_y
BY definition ,X and Y are NOT independent
: 2. Give a reasonable definition of a chi-square distribution with zero degrees
: of freedom.
: Hint: Work with the m.g.f. of a distribution that is χ^2(r) and let r = 0.
: 3. Let X be Poisson distribution with parameter m. If m is an experimental
: value of a random variable having a gamma distribution with α = 2 and
: β = 1, compute P(X = 0,1,2).
: Notation: β = 1 => λ = 1, θ = 1. They're equivalent.
M ~ gamma(2,1)
X|M=m ~ Po(m)
p(X=x)=p(X=x|M=m)*p(M=m)
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※ 編輯: mangogogo 來自: 218.170.118.39 (08/18 22:20)