ENGINEERING MATHEMATICS · PROBABILITY & STATISTICS

GATE 2020 EC, question 25: Probability & statistics — solution and answer

GATE 2020 · ECQ.25 · 1 mark · NAT Checked against the official key

The two sides of a fair coin are labelled as 0 and 1. The coin is tossed two times independently. Let $M$ and $N$ denote the labels corresponding to the outcomes of those tosses. For a random variable $X$ , defined as $X = min (M, N)$ , the expected value $E (X)$ (rounded off to two decimal places) is ________.

See it as printed in the paper

Tap to open full size

Answer and worked solution

ANSWER

$E (X) = \frac{1}{4} = 0.25$

Official key: 0.25 to 0.25 · Official answer key printed with the GATE 2020 EC paper (IIT Delhi)

Sample space for two independent tosses of a fair coin: $(M, N) \in {(0, 0), (0, 1), (1, 0), (1, 1)}$ . The coin is fair and the tosses are independent, so each outcome has probability $\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$ .
Evaluate $X = min (M, N)$ on each outcome: $min (0, 0) = 0$ , $min (0, 1) = 0$ , $min (1, 0) = 0$ , $min (1, 1) = 1$ .
So $X$ takes only the values 0 and 1, and $X = 1$ only when both tosses show 1. By independence, $P (X = 1) = P (M = 1) P (N = 1) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$ , and $P (X = 0) = 1 - \frac{1}{4} = \frac{3}{4}$ .
Definition of expectation for a discrete random variable, $E (X) = \sum x P (X = x)$ : $E (X) = 0 \cdot \frac{3}{4} + 1 \cdot \frac{1}{4} = \frac{1}{4} = 0.25$

KEY IDEA

For 0/1 variables, $min (M, N)$ equals the product $M N$ : it is 1 only when both are 1. So $X$ is Bernoulli with $p = P (M = 1) P (N = 1) = \frac{1}{4}$ , and the mean of a Bernoulli variable is $p$ . Here $E (X) = E (M N) = E (M) E (N) = \frac{1}{2} \cdot \frac{1}{2}$ , which uses the independence of the tosses.

COMMON MISTAKE

Answering 0.75 by finding $P (X = 0)$ when the question asks for $E (X)$ , or mixing up min with max: $max (M, N)$ is 1 in three of the four outcomes and gives $E = 0.75$ . A second mistake is writing $E (min (M, N)) = min (E (M), E (N)) = 0.5$ . Expectation does not commute with a nonlinear function like min.

From the GATE 2020 EC paper. Every question of this paper · Download the full paper · All 33 Probability & statistics questions, with notes

GATE 2020 EC, question 25: Probability & statistics — solution and answer

Answer and worked solution

More Probability & statistics questions