2 marksStatisticsRandom variables and standard distributionsThe normal distributionModerateabout 2 min

IIT JAM 2022 Question 31: Random variables and standard distributions

Let $X \sim N(\mu_X, \sigma_X^2)$ and $Y \sim N(\mu_Y, \sigma_Y^2)$ . Which of the following is/are NOT correct?

Select all that apply.

Answer and worked solution

Answer: (A) and (B)

Statement (A) integrates the density only up to the mean $\mu_X$ , which gives area $0.5$ , not $1$ , so it is NOT correct.

Statement (B) swaps the empirical-rule areas: $\mu \pm 2\sigma$ covers about $0.9544$ and $\mu \pm 3\sigma$ about $0.9973$ , the reverse of what is written, so it is NOT correct. Statement (C) matches the standard ratios ( $0.6745 : 0.7979 : 1 \approx \tfrac23 : \tfrac45 : 1$ ) and (D) is the correct distribution of a difference of independent normals. The NOT-correct statements are (A) and (B).

The trap: (D) If

X

and

Y

are independent, then

(X - Y) \sim N(\mu_X - \mu_Y,\ \sigma_X^2 + \sigma_Y^2)

The $\pm 2\sigma$ and $\pm 3\sigma$ coverage values are easy to transpose; keeping $0.9544$ with two sigmas and $0.9973$ with three sigmas is the check.

Plain-text version

Let X ~ N(μ_X, σ_X²) and Y ~ N(μ_Y, σ_Y²). Which of the following is/are NOT correct? (A) The area F(X) = 1/σ_X√(2π)∫_(-∞)^(μ_X) e^(-1/2((X-μ_X)/σ_X)²) dx is 1. (B) The areas under the normal curve between the ordinates at μ_X ± 3σ_X and μ_Y ± 2σ_Y are 0.9544 and 0.9973, respectively. (C) For X, Quartile Deviation: Mean Absolute Deviation: Standard Deviation ≈ 2/3σ_X: 4/5σ_X: σ_X. (D) If X and Y are independent, then (X - Y) ~ N(μ_X - μ_Y, σ_X² + σ_Y²).

Also written as: Let X ~ N(mu_X, sigma_X^2) and Y ~ N(mu_Y, sigma_Y^2). Which of the following is/are NOT correct?

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What this question tests

This question is on The normal distribution, part of Random variables and standard distributions in Statistics.

The bell curve: standardising, tail probabilities and approximations.

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