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Let $X$ be a discrete random variable (r.v) whose range is the set of non-negative integers. Let the probability mass function (PMF) of $X$ be: $PX(i)=P[X=i]=kp^i, s.t. i = 0, 1, 2, ...$ where $p \in (0,1)$ is a given parameter.

A) Find the constant k.

B) Find the conditional PMF $PX(n|X>2)$, defined as $P[X=n|X>2]$. Then find the probability that $X$ is smaller than $5$, given that $X$ is larger than $2$.

I could solve A to find that $k=(1-p)$, however I am a bit confused about B. Elaborated solutions are much appreciated.

caesar
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  • Let $A$ be the event $X=n$ and $B$ the event $X\gt 2$. We want $\Pr(A|B)$. By definition this is $\frac{\Pr(A\cap B)}{\Pr(B)}$. If $n\gt 2$, then $\Pr(A\cap B)=\Pr(X= n)$. So now all we need to finish is $\Pr(B)$. – André Nicolas Sep 02 '15 at 06:35

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