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:

GATE DA 2025 Q20: X = aZ + b with Z standard normal: find a and b

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

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

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

Try it: multiple choice: exactly one correct option

Wrong answer costs 1/3 mark in GATE.

Show answer and explanation

Answer: A (official GATE 2025 final answer key)

Concept used: Mean and covariance of a linear function of a standard normal variable.

First step: Since $E[Z] = 0$ , $E[X] = b$ . Then $X - E[X] = aZ$ , so $E[(X - E[X])Z] = a\,E[Z^2] = a$ .

Detailed working, common mistakes and linked mock tests will be available with a free GATEiit account. Log-in is coming soon.

Share this question

Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key