GATE 2025 DA – Question 20

Probability and Statistics · Mean, median, mode, standard deviation, correlation and covariance · 1 mark · Multiple choice

Let $X = aZ + b$, where $Z$ is a standard normal random variable, and $a$, $b$ are two unknown constants. It is given that

$E[X] = 1$, $E[(X - E[X])Z] = -2$, $E[(X - E[X])^2] = 4$,

where $E[X]$ denotes the expectation of random variable $X$. The values of $a$, $b$ are:

$a = -2$, $b = 1$
$a = 2$, $b = -1$
$a = -2$, $b = -1$
$a = 1$, $b = 1$

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Correct answer: (A) $a = -2$, $b = 1$

Explanation

Since $E[Z] = 0$, $E[X] = b = 1$. Next, $X - E[X] = aZ$, so $E[(X - E[X])Z] = aE[Z^2] = a = -2$. As a check, $E[(X - E[X])^2] = a^2E[Z^2] = 4$, which agrees. So $a = -2$ and $b = 1$.