Let $X$ be a geometric random variable and let $A$ denote the event $\{X>3\}$ find the conditional probability mass function of $X$ with respect to the event $A$ and then compute $E[X\mid A]$.
Conditional expectation of a Geometric random variable conditioned on the event A=\{X>3\}
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probability
1 Answers
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We have that $ P(X=k)=p(1-p)^k,\quad k=0,1,2,\ldots $ and hence $P(X>3)=(1-p)^4$. Now if $k=0,1,2,3$, then of course $P(X=k\mid X>3)=0$ and if $k=4,5,\ldots$, then $ P(X=k\mid X>3)=\frac{P(X=k,X>3)}{P(X>3)}=\frac{P(X=k)}{P(X>3)}=p(1-p)^{k-4}. $ The conditional expectation is thus given by $ E[X\mid X>3]=\sum_{k=4}^\infty kP(X=k\mid X>3)=\sum_{k=4}^\infty kp(1-p)^{k-4}=\sum_{k=0}^\infty (k+4)p(1-p)^k=4+E[X]. $
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0Thank you! This really helped, hopefully I remember it on my final tomorrow! – 2012-12-14