Question

Let $X$ and $Y$ be independent non negative integer valued random variables with $E(X) < \infty$, $E(Y) < \infty$, then

• A. $E(\min(X, Y)) = \sum_{R=0}^{\infty} P(XY > R)$
• B. $E(\min(X, Y)) = \sum_{R=0}^{\infty} P(X \le R)P(Y \le R)$
• C. $E(\min(X, Y)) = \sum_{R=0}^{\infty} P(X > R) \cdot P(Y > R)$
• D. $E(\min(X, Y)) = \sum_{R=0}^{\infty} P(X Y \le R)$

Hint:

The discrete version of the survival function expectation rule is $\sum P(X > n)$, while the continuous version is $\int P(X > x) dx$. They are equivalent representations of the same fundamental concept.

Answer 1

The Correct Option is C

For discrete non-negative integer-valued random variables, there is a standard summation formula for the expected value.

Step 1: \color{redExpectation Formula for Discrete Non-negative Variables
For a discrete random variable $Z$ taking values in $\{0, 1, 2, \dots\}$, the expectation is given by:
$E(Z) = \sum_{R=0}^{\infty} P(Z > R)$

Step 2: \color{redApplying the Formula to the Minimum
Let $Z = \min(X, Y)$.
Then $E(\min(X, Y)) = \sum_{R=0}^{\infty} P(\min(X, Y) > R)$

Step 3: \color{redUtilizing Independence
The event $(\min(X, Y) > R)$ is equivalent to both $X > R$ and $Y > R$.
$P(\min(X, Y) > R) = P(X > R \cap Y > R)$
Since $X$ and $Y$ are independent:
$P(X > R \cap Y > R) = P(X > R) \cdot P(Y > R)$

Step 4: \color{redConstructing the Final Sum
Substitute this back into the expectation formula:
$E(\min(X, Y)) = \sum_{R=0}^{\infty} P(X > R) \cdot P(Y > R)$
This matches Option (3).