GATE 2024 EC Q.33: solution and answer

GATE 2024 · ECQ.33 · 1 mark · NAT

Suppose $X$ and $Y$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0, 1]$ . The probability that $X \geq Y$ is ________.

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Answer and solution

ANSWER

$P (X \geq Y) = \frac{1}{2} = 0.5$

Official answer: 0.50Source: Official answer key, GATE 2024 EC (IISc Bengaluru)

Because $X$ and $Y$ are independent, the joint density is the product of the two marginals: $f_{X, Y} (x, y) = f_{X} (x) f_{Y} (y) = 1 \cdot 1 = 1$ on the unit square $0 \leq x \leq 1, 0 \leq y \leq 1$ , and $0$ outside it.
The joint density is constant at 1, so the probability of an event is the area of its region inside the unit square: $P (X \geq Y) = \iint_{x \geq y} 1 d x d y$ .
The region $x \geq y$ inside the square is the triangle on or below the diagonal $y = x$ , with vertices $(0, 0)$ , $(1, 0)$ and $(1, 1)$ .
Integrate over it: $P (X \geq Y) = \int_{0}^{1} \int_{0}^{x} 1 d y d x = \int_{0}^{1} x d x = \frac{1}{2} .$ This is also the triangle's area, $\frac{1}{2} \times 1 \times 1$ .
Check by symmetry: $X$ and $Y$ are i.i.d., so $(X, Y)$ and $(Y, X)$ have the same distribution and $P (X \geq Y) = P (Y \geq X)$ . The variables are continuous, so $P (X = Y) = 0$ (the diagonal has zero area). The two events therefore cover the whole square and overlap only on a set of probability zero, so their probabilities add to 1 and each equals $\frac{1}{2}$ .
So $P (X \geq Y) = 0.5$ .

KEY IDEA

For independent uniform variables on $[0, 1]$ the joint density is 1 on the unit square, so any probability equals the area of the event region. The region $x \geq y$ is half the square. Symmetry gives the same answer with no integration: for i.i.d. continuous variables, $P (X \geq Y) = P (Y \geq X)$ and $P (X = Y) = 0$ .

WATCH OUT FOR

Some students think the equality case $X = Y$ needs extra probability and give an answer above 0.5. It does not: for continuous variables $P (X = Y) = 0$ , so $P (X \geq Y) = P (X > Y) = 0.5$ . Another slip is setting the inner integral limits the wrong way round (for example $\int_{x}^{1} d y$ ), which describes the region $y \geq x$ . Here that still gives 0.5 by symmetry, but the same slip gives a wrong answer when the variables are not identically distributed.

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GATE 2024 EC Q.33: solution and answer

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