GATE 2024 DA – Question 56
Let $X$ be a random variable uniformly distributed in the interval $[1, 3]$ and $Y$ be a random variable uniformly distributed in the interval $[2, 4]$. If $X$ and $Y$ are independent of each other, the probability $P(X \ge Y)$ is ______ (rounded off to three decimal places).
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Correct answer: 0.115 to 0.135
Explanation
The densities are $\frac{1}{2}$ each. $X \ge Y$ is possible only when $X > 2$, as $Y \ge 2$. For $x \in [2, 3]$, $P(Y \le x) = \frac{x - 2}{2}$. So $P(X \ge Y) = \int_2^3 \frac{1}{2}\cdot\frac{x - 2}{2}dx = \frac{1}{4}\cdot\frac{1}{2} = \frac{1}{8} = 0.125$.