Recent questions in Probability and Statistics

Suppose that a random variable $X$ has a probability density function\[\begin{aligned}f(x) & =c(x-4) \quad \text { for } 4 \leq x \leq 6 \\& =0 \quad \text { for all othe...

A fair dice (with faces numbered $1, \ldots, 6$ ) is independently rolled twice. Let $X$ denote the maximum of the two outcomes. The expected value of $X$ is$4 \frac{1}{2...

Let $X$ be a Gaussian random variable with mean $\mu_{1}$ and variance $\sigma_{1}^{2}$. Now, suppose that $\mu_{1}$ itself is a random variable, which is also Gaussian d...

A non-negative loss in a car accident is distributed with the following probability density function\[f(x)=\frac{1}{10} \exp (-x / 10)\]for $x \geq 0$. Suppose that first...

$X$ and $Y$ are jointly Gaussian random variables with zero mean.A constant-pdf contour is where the joint density function takes on the same value. If the constant-pdf c...

Consider a fair coin that has probability $1 / 2$ of showing heads $(\text{H})$ in a toss and $1 / 2$ of showing tails $(\text{T})$. Suppose we independently flip a fair ...

Let $X$ and $Y$ be two zero mean independent continuous random variables. Let $Z_{1}=\max (X, Y)$, and $Z_{2}=\min (X, Y)$. Then which of the following is TRUE.$Z_{1}$ an...

$X, Y, Z$ are integer valued random variables with the following two properties:$X$ and $Y$ are independent.For all integer $x$, conditioned on the event $\{X=x\}$, we ha...

A surprise quiz contains three multiple choice questions; question $1$ has $3$ suggested answers, question $2$ has four, and question $3$ has two. A completely unprepared...

Consider four coins, three of which are fair, that is they have heads on one side and tails on the other and both are equally likely to occur in a toss. The fourth coin h...

Consider a coin tossing game between Santa and Banta. Both of them toss two coins sequentially, first Santa tosses a coin then Banta and so on. Santa tosses a fair coin: ...

Suppose three dice are rolled independently. Each dice can take values $1$ to $6$ with equal probability. Find the probability that the second highest outcome equals the ...

A Poisson random variable $X$ is given by $\operatorname{Pr}\{X=k\}=\mathrm{e}^{-\lambda} \lambda^{k} / k !, k=0,1,2, \ldots$ for $\lambda>0$. The variance of $X$ scales ...

In modeling the number of health insurance claims filed by an individual during a three year period, an analyst makes a simplifying assumption that for all non-negative i...

Consider a single amoeba that at each time slot splits into two with probability $p$ or dies otherwise with probability $1-p$. This process is repeated independently infi...

Let $X$ and $Y$ be indepedent, identically distributed standard normal random variables, i.e., the probability density function of $X$ is\[f_{X}(x)=\frac{1}{\sqrt{2 \pi}}...

Consider a string of length $1 \mathrm{~m}$. Two points are chosen independently and uniformly random on it thereby dividing the string into three parts. What is the prob...

Consider two independent random variables $X$ and $Y$ having probability density functions uniform in the interval $[-1,1]$. The probability that $X^{2}+Y^{2}>1$ is$\pi /...

What is the value of $\lambda$ such that $\operatorname{Prob}\{X>\operatorname{mean}\{X\}\}=1 / e$, where $\text{PDF}$ of $X$ is $p_{X}(x)=\lambda e^{-\lambda x}, x \geq ...

Consider two independent random variables $\text{X}$ and $\text{Y}$ having probability density functions uniform in the interval $[0,1]$. When $\alpha \geq 1$, the probab...