GATE 2026 Engineering Mathematics PYQs | PROBABILITY & STATISTICS

Let X be a random variable which takes values in the set {1, 2, 3, 4, 5, 6, 7, 8}. Further,

Pr(X = 1) = Pr(X = 2) = Pr(X = 5) = Pr(X = 7) = 1/6
Pr(X = 3) = Pr(X = 4) = Pr(X = 6) = Pr(X = 8) = 1/12

The expected value of X, denoted by E[X], is equal to ____. (Round off to two decimal places) [NAT || GATE-2026 Set-1 || 2-marks]

An urn contains one red ball and one blue ball. At each step, a ball is picked uniformly at random from the urn, and this ball together with another ball of the same color is put back in the urn. The probability that there are equal number of red and blue balls after two steps is [MCQ || GATE-2026 Set-1 || 2-marks]

[Tex]\begin{aligned} & \text{The probability density function } f(x) \text{ of a random} \\ & \text{variable } X \text{ which takes real values is} \\ & \qquad\qquad f(x) = \frac{1}{3\sqrt{2\pi}} \exp\left(-\frac{x^2}{18}\right), \; x \in (-\infty, +\infty) \\ & \text{Which one of the following statements is correct} \\ & \text{about the random variable } X \text{ ?} \textbf{[MCQ || GATE-2026 Set-2 || 1-marks]} \end{aligned}[/Tex]

Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all previous coin tosses. Let E1 be the event that among the second, fourth, and sixth coin tosses, there are at least two heads. Let E2 be the event that among the first, second, third, and fifth coin tosses, there are equal number of heads and tails. The conditional probability P(E1 | E2) is equal to the _____.(rounded off to one decimal place) [NAT || Gate-2026 Set-2 || 2-marks]