Expected Value of $Z = min(X, M)$ where M is a constant

Question

Let $X \sim Geometric(p)$, and let $M > 0$ be a positive integer. Determine the expected value of $Z = min(X, M)$.

I have done similar exercises where $M$ is a random variable, but not a constant. I don't know how to proceed here, and therefore have no work to show. Hints are very welcome although It would be nice to see more or less of a complete solution for a case like this.

Thank you.

Simply calculate $E(Z)=\sum_j \min(j,M)p(j)$, where $p$ is the pmf of $X$. — StubbornAtom, May 14 '19 at 14:18
$\min(j,M)=j$ if $j<M$ and equals $M$ if $j\ge M$. So break the sum accordingly. — StubbornAtom, May 14 '19 at 14:27

score 1 · Answer 1 · answered May 14 '19 at 15:05

Note that $$ EZ=EXI(X<M)+EMI(X\geq M)\tag{0} $$ where $I$ is the indicator function since $Z=X$ if $X<M$ and $Z=M$ if $X\geq M$. But $$ EMI(X\geq M)=MP(X\geq M)\tag{1} $$ and $$ EXI(X<M)=\sum_{k=0}^{M-1}kP(X=k).\tag{2} $$ Since $X$ is geometric we can compute, $(1)$ and $(2)$ and hence $(0)$. I leave this computation to you.

Expected Value of $Z = min(X, M)$ where M is a constant

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