Question
For a nonnegative integer-valued random variable $N$, show that$$E[N]=\sum_{i=1}^{\infty} P\{N \geq i\}$$
Step 1
Step 1: First, we need to understand that the expectation of a random variable $N$ is defined as $E[N] = \sum_{i=0}^{\infty} i \cdot P\{N = i\}$. Show more…
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Key Concepts
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For a nonnegative integer-valued random variable $N$, show that $$ \sum_{i=0}^{x} i P[N>i\}=\frac{1}{2}\left(E\left[N^{2}\right]-E[N]\right) $$
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Theoretical Exercises
Show that if $X$ is a discrete random variable, taking values on the positive integers, then $E(X)=\sum_{k=1}^{\infty} P(X \geq k) .$ Apply this result to find the expected value of a geometric random variable.
If $X$ is a nonnegative integer valued random variable, show that (a) $$ E[X]=\sum_{n=1}^{\infty} P[X \geq n\}=\sum_{n=0}^{\infty} P(X>n\} $$ Hint: Define the sequence of random variables $I_{n}, n \geq 1$, by $$ I_{n}=\left\{\begin{array}{ll} 1, & \text { if } n \leq X \\ 0, & \text { if } n>X \end{array}\right. $$ Now express $X$ in terms of the $I_{n}$. (b) If $X$ and $Y$ are both nonnegative integer valued random variables, show that $$ E[X Y]=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} P(X \geq n, Y \geq m) $$
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