GATE CS 1998 Q1.1 — Probability: Random Variables and Distributions

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

Let $X$ be a random variable which takes values in the set $\{1,2,3,4,5,6,7,8\}$. Further, $\operatorname{Pr}(X=1)=\operatorname{Pr}(X=2)=\operatorname{Pr}(X=5)=\operatorname{Pr}(X=7)=\frac{1}{6}$ and $\operatorname{Pr}(X=3)=\operatorname{Pr}(X=4)=\operatorname{Pr}(X=6)=\operatorname{Pr}(X=8)=\frac{1}{12}$. The expected value of $X$, denoted by …

The probability density function $f(x)$ of a random variable $X$ which takes real values is $$f(x)=\frac{1}{3 \sqrt{2 \pi}} \exp \left(-\frac{x^{2}}{18}\right), \quad x \in(-\infty,+\infty)$$ Which one of the following statements is correct about the random variable $X$ ?

A box contains $5$ coins: $4$ regular coins and $1$ fake coin. When a regular coin is tossed, the probability $P($ head $)=0.5$ and for a fake coin, $P($ head $)=1$. You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is __________. (rounded off to two decimal places)

Consider a probability distribution given by the density function $P(x)$. $$P(x)=\left\{\begin{aligned} C x^{2}, & \text{ for } 1 \leq x \leq 4 \\ 0, & \text{ for } x<1 \text{ or } x>4 \end{aligned}\right.$$ The probability that $x$ lies between $2$ and $3$, i.e., $P(2 \leq x \leq 3)$ is _________. (rounded off to three decimal …

A quadratic polynomial $(x-\alpha)(x-\beta)$ over complex numbers is said to be square invariant if $(x-\alpha)(x-\beta)=\left(x-\alpha^{2}\right)\left(x-\beta^{2}\right)$. Suppose from the set of all square invariant quadratic polynomials we choose one at random. The probability that the roots of the chosen polynomial are equal is __________. (rounded …