※ 引述《wisely13 (two-6)》之銘言：
: 7.If Y is geometric random variable with E(Y)=7/3, determine
: (a) Pr(Y=3)
: (b) Pr(Y>=3)
: (c) Pr(Y>=5)
: (d) Pr(Y>=5 | Y>=3)
: (e) Pr(Y>=6 | Y>=4)
: (f) Var(Y)
設成功機率為p
E(Y)=1/p
p=3/7
: (a) Pr(Y=3)
Pr(Y=3)=[(4/7)^2]*(3/7)=(16/49)*(3/7)=48/343
: (b) Pr(Y>=3)
∞
Pr(Y>=k)=Σ [(1-p)^(x-1)]*p = [(1-p)^(k-1)]*p/[1-(1-p)] = (1-p)^(k-1)
x=k
Pr(Y>=3)=(1-3/7)^(3-1) = (4/7)^2 = 16/49
: (c) Pr(Y>=5)
Pr(Y>=5)=(1-3/7)^(5-1) = (4/7)^4 = 256/2401
: (d) Pr(Y>=5 | Y>=3)
Pr(Y>=5且Y>=3) Pr(Y>=5) 256/2401
Pr(Y>=5 | Y>=3)= ─────── = ─────── = ───── = 16/49
Pr(Y>=3) Pr(Y>=3) 16/49
: (e) Pr(Y>=6 | Y>=4)
Pr(Y>=6且Y>=4) Pr(Y>=6) (1-3/7)^(6-1) (4/7)^5
Pr(Y>=6 | Y>=4)= ─────── = ─────── = ─────── = ────
Pr(Y>=4) Pr(Y>=4) (1-3/7)^(4-1) (4/7)^3
= (4/7)^2 = 16/49
(可用無記憶性質來解題)
: (f) Var(Y)
Var(Y)=(1-p)/(p^2) = (4/7)/[(3/7)^2]=(4/7)/(9/49)= 28/9
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