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13 votes
13 votes

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.
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Two random variables $X$ and $Y$ are said to be Independent and Identically distributed (ie. IID) iff

{\displaystyle {\begin{aligned}&F_{X}(x)=F_{Y}(x)\,&\forall x\in I\\&F_{X,Y}(x,y)=F_{X}(x)\cdot F_{Y}(y)\,&\forall x,y\in I\end{aligned}}}

$F_X$ and $F_Y$ is CDF of Random variable $X$ and $Y$ respectively. 

Ref: wiki source

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1

1 Answer

11 votes
11 votes
Best answer

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.

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4 Comments

What is the meaning of ' identically distributed random variables'?
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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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0
Answer:

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