GATE 2026 DA Q.44: solution and answer

GATE 2026 · DAQ.44 · 2 marks · MCQ

Let $X$ and $Y$ be two independent random variables. $X$ follows $B er n o u l l i (p = 0.3)$ distribution and $Y$ follows $N or ma l (μ = 0, σ^{2} = 100)$ distribution.

Which of the following options is the variance of $(2 X - 1) Y$ ?

(A)
100
(B)
90
(C)
49
(D)
21

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

The variance of $(2 X - 1) Y$ is $100$ , which is option (A).

Official answer: ASource: Official answer key, GATE 2026 DA (IIT Guwahati)

Let $W = 2 X - 1$ . Since $X \in {0, 1}$ , $W = - 1$ when $X = 0$ and $W = 1$ when $X = 1$ . So $W^{2} = 1$ with certainty, and $E [W^{2}] = 1$ whatever the value of $p$ .
$W$ is a function of $X$ alone and $X$ is independent of $Y$ , so $W$ and $Y$ are independent.
Because $W$ and $Y$ are independent and $E [Y] = 0$ : $E [W Y] = E [W] E [Y] = 0$ .
$W^{2}$ and $Y^{2}$ are also independent, so $E [W^{2} Y^{2}] = E [W^{2}] E [Y^{2}]$ .
Hence $Var (W Y) = E [W^{2} Y^{2}] - (E [W Y])^{2} = E [W^{2}] E [Y^{2}] - 0$ .
$E [Y^{2}] = Var (Y) + (E [Y])^{2} = 100 + 0 = 100$ , so $Var ((2 X - 1) Y) = 1 \times 100 = 100$ .
Check with the general formula for independent $W$ and $Y$ : $Var (W Y) = E [W^{2}] σ_{Y}^{2} + Var (W) μ_{Y}^{2}$ . Here $E [W] = 2 (0.3) - 1 = - 0.4$ and $Var (W) = 4 \cdot 0.3 \cdot 0.7 = 0.84$ , so the value is $1 \cdot 100 + 0.84 \cdot 0 = 100$ . The answer is option (A).

KEY IDEA

For independent $W$ and $Y$ with $E [Y] = 0$ , $Var (W Y) = E [W^{2}] E [Y^{2}]$ . Here $2 X - 1$ takes only the values $\pm 1$ , so $E [W^{2}] = 1$ and the variance equals $Var (Y)$ , whatever $p$ is.

WATCH OUT FOR

Using $Var (2 X - 1) Var (Y) = 0.84 \times 100 = 84$ , which wrongly treats the variance of a product as the product of the variances. Another slip is using $4 p (1 - p) \cdot 100$ and ignoring that $E [(2 X - 1)^{2}] = 1$ , not $0.84$ .

From the GATE 2026 DA paper. Every question of this paper · Download the full paper · All 8 Random variables questions, with notes

GATE 2026 DA Q.44: solution and answer

Answer and solution

More Random variables questions