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Let $X$ and $Y$ be two independent and identically distributed random variables. Then $P\left ( X> Y \right )$ is.

1. $\frac{1}{2}$
2. 1
3. 0
4. $\frac{1}{3}$
5. Information is insufficient.

### 1 comment

Two random variables $X$ and $Y$ are said to be Independent and Identically distributed (ie. IID) iff $F_X$ and $F_Y$ is CDF of Random variable $X$ and $Y$ respectively.

Ref: wiki source

Let the probability $P({X = Y}) > 0$. This can happen if $X$ and $Y$ are discrete random variables. Also, if $X$ and $Y$ are continuous random variables, it could be that some values have a non-zero probability of getting selected.

Then $P({X>Y}) = P({Y>X}) = \dfrac{1 - P({X=Y})}{2}$

Since nothing is said about the value of $P({X=Y})$, the correct answer will be option e. Information is insufficient.

What is the meaning of ' identically distributed random variables'?
Why it is not 1/3, as there are 3 possibilities for X and Y (X>Y, X<Y, X=Y)

So probability of X>Y will be 1/3.

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