GATE Overflow for GATE XE - Recent activity in Probability and Statistics

Edited: GATE XE 2026 | Question: 7

Sat, 28 Mar 2026 10:00:49 +0000

Let $X$ be a random variable having probability density function (pdf)

$$f_X(x) =
\left\{
\begin{array}{ll}
\dfrac{e}{2(e-1)} e^{-x}, & 0 < x < 1, \\
\dfrac{1}{2}, & 1 < x < 2, \\
0, & \text{elsewhere}.
\end{array}
\right.$$

Then, the value of the expectation of $X, E[X]$ (rounded off up to two decimal places) is $\_\_\_\_$

Edited: GATE XE 2026 | Question: 4

Sat, 28 Mar 2026 09:58:30 +0000

Which of the following statements is/are true?

$Y=1+1.8 X$ and $X=1-0.5 Y$ are estimated as lines of regression of $Y$ on $X$ and $X$ on $Y$, respectively.
$Y=1+1.3 X$ and $X=2+1.1 Y$ are estimated as lines of regression of $Y$ on $X$ and $X$ on $Y$, respectively.
$Y=1+1.6 X$ and $X=3+0.5 Y$ are estimated as lines of regression of $Y$ on $X$ and $X$ on $Y$, respectively.
$Y=1+X$ and $X=Y$ are estimated as lines of regression of $Y$ on $X$ and $X$ on $Y$, respectively, and the correlation between $X$ and $Y$ is $\pm 1$.

Edited: GATE XE 2025 | Question: 11

Sun, 29 Jun 2025 11:16:05 +0000

Let $X$ and $Y$ be two random variables with mean $0$, variance $1$, and correlation coefficient $\frac{1}{3}$. Then the value of $\operatorname{Var}(X+3 Y)$ is equal to

$9$
$10$
$11$
$12$

Recategorized: GATE XE 2024 | Question: 3

Mon, 05 May 2025 14:41:33 +0000

There are four cities, namely, $C_{1}, C_{2}, C_{3}$ and $C_{4}$. The cities are directly connected by four roads as shown in the picture given below, that is, $C_{1}$ is connected with $C_{2}$, $C_{2}$ is connected with $C_{3}, C_{3}$ is connected with $C_{4}$, and $C_{4}$ is connected with $C_{1}$. The probability of any road getting independently blocked is $\frac{1}{3}$. Let $E_{1}$ be the event of travelling from $C_{1}$ to $C_{3}$ via $C_{2}$ and $E_{2}$ be the event of travelling from $C_{1}$ to $C_{3}$ via $C_{4}$. Then, which of the following statements is correct ?

$\quad P\left(E_{1} \cup E_{2}\right)=\frac{56}{81}$
$\quad P\left(E_{1} \cup E_{2}\right)=\frac{8}{9}$
$\quad P\left(E_{1} \mid E_{2}\right) \neq P\left(E_{1}\right)=\frac{4}{9}$
$\quad P\left(E_{1} \cap E_{2}\right)=0$

Recategorized: GATE XE 2022 | Question: 7

Mon, 05 May 2025 12:44:08 +0000

In a cosmopolitan city, the population comprises of $30 \%$ female and $70 \%$ male. Suppose that $5 \%$ of female and $30 \%$ of male in the population are foreigners. A person is selected at random from this population. Given that the selected person is a foreigner, the probability that the person is a female is three decimal places).

Recategorized: GATE XE 2022 | GA Question: 4

Mon, 05 May 2025 12:41:33 +0000

A survey of $450$ students about their subjects of interest resulted in the following outcome.

- $150$ students are interested in Mathematics.

- $200$ students are interested in Physics.

- $175$ students are interested in Chemistry.

- $50$ students are interested in Mathematics and Physics.

- $60$ students are interested in Physics and Chemistry.

- $40$ students are interested in Mathematics and Chemistry.

- $30$ students are interested in Mathematics, Physics and Chemistry.

- Remaining students are interested in Humanities.

Based on the above information, the number of students interested in Humanities is

$10$
$30$
$40$
$45$

Recategorized: GATE XE 2023 | Question: 10

Mon, 05 May 2025 12:21:57 +0000

The probability of a person telling the truth is $\frac{4}{6}$. An unbiased die is thrown by the same person twice and the person reports that the numbers appeared in both the throws are same. Then the probability that actually the numbers appeared in both the throws are same is ________$\text{(round off to 2 decimal places)}$.