Recent questions tagged probability-and-statistics

Let $\text{X}$ and $\text{Y}$ be two real-valued random variables with $\mathrm{E}(\mathrm{X})=1, \mathrm{E}(\mathrm{Y})=2, \mathrm{E}\left(\mathrm{X}^{2}\right)=4, \math...

Consider discrete random variables $X$ and $Y$ with probabilities as follows:$$ \begin{array}{l} P(X=0 \text { and } Y=0)=\frac{1}{4} \\ P(X=1 \text { and } Y=0)=\frac{1}...

Let $X$ and $Y$ be continuous random variables with probability density functions $P_{X}(x)$ and $P_{Y}(y)$, respectively. Further, let $Y=X^{2}$ and$$P_{X}(x)=\begin{cas...

​​​​​Let $\text{X}$ be a discrete random variable that is uniformly distributed over the set $\{-10,-9, \cdots, 0, \cdots, 9,10\}$. Which of the following random variable...

One million random numbers are generated from a statistically stationary process with a Gaussian distribution with mean zero and standard deviation $\sigma_o$.The $\sigma...

Let the probability density function of a random variable $x$ be given as $$f\left ( x \right ) = ae^{-2\left | x \right |}$$ The value of $’a’$ is _________________.

Suppose the probability that a coin toss shows "head" is $p$, where $0<p<1$. The coin is tossed repeatedly until the first "head" appears. The expected number of tosses r...

The mean-square of a zero-mean random process is $\frac{kT}{c},$ where $k$ is Boltzmann’s constant, $T$ is the absolute temperature, and $C$ is a capacitance. The standar...

The probability of a resistor being deflective is $0.02$. There are $50$ such resistors in a circuit. The probability of two or more defective resistors in the circuit (r...

$A$ and $B$ are friends. They decide to meet between $1:00$ pm and $2:00$ pm on a given day. There is a condition that whoever arrives first will not wait for the other f...

Two independent random variables $X$ and $Y$ are uniformly distributed in the interval $[-1,1]$. The probability that $\text{max}[X,Y]$ is less than $1/2$ is$3/4$$9/16$$1...

There are $3$ red socks, $4$ green socks and $3$ blue socks. You choose $2$ socks. The probability that they are of the same colour is$1/5$$7/30$$1/4$$4/15$

A person decides to toss a fair coin repeatedly until he gets a head. He will make at most $3$ tosses. Let the random variable $Y$ denote the number of heads. The value o...

Assume that in traffic junction, the cycle of the traffic signal lights is $2$ minutes of green (Vehicle does not stop) and $3$ minutes of red (Vehicle stops). Consider t...

The probability that a $k$-digit number does NOT contain the digits $0, 5$ or $9$ is$0.3^{k}$$0.6^{k}$$0.7^{k}$$0.9^{k}$

What is the chance that a leap year, selected at random, will contain $53$ Saturdays?$2/7$$3/7$$1/7$$5/7$

A continuous random variable $X$ has a probability density function $f(x)=e^{-x}, 0< x< \infty$. then $P\{X 1\}$$0.368$$0.5$$0.632$$1.0$

Lifetime of an electric bulb is a random variable with density $f(x)=kx^2$ , where $x$ is measured in years. If the minimum and maximum lifetimes of bulb are $1$ and $2$ ...

Let $X$ be a random variable with probability density function$f(x)=\begin{cases} 0.2,& \text{for } \mid x \mid \leq 1\\ 0.1,& \text{for }1< \mid x \mid \leq 4\\ 0 & \tex...

A fair coin is tossed $n$ times. The probability that the difference between the number of heads and tails is $(n-3)$ is$2^{-n}$$0$$^{n}C_{n-3}2^{-n}$$2^{-n+3}$

Two coins $R$ and $S$ are tossed. The $4$ joint events $H_{R}H_{S}, T_{R}T_{S}, H_{R}T_{S}, T_{R}H_{S}$ have probabilities $0.28, 0.18, 0.30, 0.24$, respectively, where $...