2 marks, −0.667StatisticsRandom variables and standard distributionsTransformations and order statisticsModerateabout 2 min

IIT JAM 2021 Question 23: Random variables and standard distributions

$X$ and $Y$ are independent with $F_X(x) = 1 - \left(\tfrac34\right)^x$ and $G_Y(y) = 1 - \left(\tfrac23\right)^y$ for $x, y = 1, 2, 3, \ldots$ . Let $Z = \min\{X, Y\}$ . Then $P(Z \ge 6)$ is

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Answer and worked solution

Answer: (B)

From the CDFs, $P(X \ge 6) = 1 - F_X(5) = \left(\tfrac34\right)^5$ and $P(Y \ge 6) = \left(\tfrac23\right)^5$ .

By independence, $P(Z \ge 6) = P(X \ge 6)\,P(Y \ge 6) = \left(\tfrac34 \cdot \tfrac23\right)^5 = \left(\tfrac12\right)^5 = \tfrac{1}{32}$ . The answer is (B).

The trap: (A)

\dfrac{1}{64}

$Z \ge 6$ needs both $X$ and $Y$ at least 6, so the two tail probabilities multiply; using a single power of $\tfrac12$ too many or too few gives $\tfrac{1}{64}$ .

Plain-text version

X and Y are independent with F_X(x) = 1 - (3/4)^x and G_Y(y) = 1 - (2/3)^y for x, y = 1, 2, 3, …. Let Z = min {X, Y}. Then P(Z ≥ 6) is (A) 1/64 (B) 1/32 (C) 63/64 (D) 31/32

Also written as: X and Y are independent with F_X(x) = 1 - (3/4)^x and G_Y(y) = 1 - (2/3)^y for x, y = 1, 2, 3,.... Let Z = min {X, Y}. Then P(Z >= 6) is

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What this question tests

This question is on Transformations and order statistics, part of Random variables and standard distributions in Statistics.

The distribution of a function of a random variable, and of the largest or smallest of several.

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