Recent questions tagged goclasses-statistics-practice-questions

Let $X$ and $Y$ be two random variables with means $\mu_X=2, \mu_Y=3$, variances $\sigma_X^2=4, \sigma_Y^2=9$, and covariance $\operatorname{Cov}(X, Y)=3$. Define new ran...

Let $\left\{X_i\right\}$ be a sequence of independent and identically distributed (i.i.d.) random variables with $E\left[X_i\right]=\mu$ and $\operatorname{Var}\left(X_i\...

Let $X_1$ and $X_2$ be two random variables with $\operatorname{Var}\left(X_1\right)=4, \operatorname{Var}\left(X_2\right)=9$, and $\operatorname{Cov}\left(X_1, X_2\right...

Let $X$ be a random variable with $E[X]=2$ and $\operatorname{Var}(X)=1$. Define a new random variable $Y= 2 X-X^2$.The expected value $E[Y]$ is:$-2$ $-1$ $0$ $2$

$\text { Let } X \text { and } Y \text { be two random variables with the joint probability mass function (pmf) given by: }$\begin{array}{} & Y=-1 & Y=0 & Y=1 \\X=0 & 0.1...

A box contains $\mathbf{6}$ cards numbered $\mathbf{1}$ to $\mathbf{6}$ .You draw two cards without replacement. Let the first drawn number be $A$ and the second be $B$.D...

A random variable $N$ takes values $2,3,4$ with probabilities $1 / 4,~1 / 2,~1 / 4$ respectively. Given $N=n$, another random variable $Y$ is $\operatorname{Binomial}(n, ...

A box contains $\textbf{8}$ balls numbered $\textbf{1}$ to $\textbf{8}$.You draw three balls without replacement.Let the drawn numbers be $A, B, C$.Define$$X=\min (A, B, ...

A box contains $4$ red balls and $6$ blue balls.Two balls are drawn one after another without replacement.Define a random variable$X=(1$ if the two balls are of the same ...

A fair die is rolled twice. Let $X$ be the maximum of the two outcomes and $Y$ be the minimum of the two outcomes.Define$Z=3 X-2 Y.$Find $\mathrm{E}(\mathrm{Z})$.

A shopkeeper sells an item for ₹$100$. The probability that a customer buys the item on any given day is $0.3$. If a customer buys it, the profit is ₹$20$; otherwise, the...

The joint distribution of two discrete random variables $X$ and $Y$ is given as follows:\[\begin{array}{|c|c|c|}\hlineX & Y & P(X, Y) \\\hline0 & 0 & 0.2 \\\hline0 & 1 & ...

The probability distribution of a discrete random variable $X$ is given by\[\begin{array}{c|cccc}X & 0 & 1 & 2 & 3 \\\hlineP(X) & 0.1 & 0.3 & 0.4 & 0.2\end{array}\] Find ...

A dietitian claims that a new diet plan reduces average weight by $\mathbf{5~kg}$ in a month.A random sample of $\mathbf{9}$ individuals on this diet shows a mean reducti...

A manufacturer claims that the average lifetime of its LED bulbs is $\mathbf{1000}$ hours.A consumer agency randomly tests $\mathbf{50}$ bulbs and finds their mean lifeti...

A random variable $X$ takes values $1, 2, 3$ with equal probability.Define another random variable $Y=X^2+1$.Find the covariance between $X$ and $Y$.

Two random variables $X$ and $Y$ are jointly distributed such that$$E[X]=2, \quad E[Y]=3, \quad \operatorname{Var}(X)=1, \quad \operatorname{Var}(Y)=4, \quad \text { and ...

A random variable $X$ is defined as follows:$$X= \begin{cases}U_1, & \text { with probability } \frac{1}{2} \\\\ U_2, & \text { with probability } \frac{1}{2}\end{cases}$...

A box contains $4$ red balls and $6$ blue balls. Two balls are drawn without replacement.Let random variable $X$ denote the number of red balls drawn.Find $E\left[X^2\rig...

Let $N$ be a random variable with$$P(N=n)=\frac{1}{3}\left(\frac{2}{3}\right)^n, \quad n=0,1,2, \ldots$$Given $N=n$, let $X_1, X_2, \ldots, X_n$ be i.i.d. random variable...

Let $X$ and $Y$ be random variables such that$$E[X]=2, \quad E[Y]=3, \quad \operatorname{Var}(X)=4, \quad \operatorname{Var}(Y)=9,$$and their correlation coefficient is $...

Let $X$ and $Y$ be independent Bernoulli random variables with $P(X=1)=p$ and $P(Y=1)=q$.Define $Z=X+Y$.Find Var$(\mathbf{Z})$ in terms of $p$ and $q$. $p+q-2 p q$ $p(1-p...

A fair six-sided die is rolled once.Let the random variable $X=2 Y+1$, where $Y$ is the number shown on the die. Find $\operatorname{Var}(\mathrm{X})$. $35 / 12$ $35 / 3$...

Let $X$ be a discrete random variable taking values $0,1,2, \ldots$ with $P(X=k)=\frac{1}{(k+1)(k+2)}$ for $k=0,1,2, \ldots$.Find $\mathbb{E}[X]$. $1$ $2$ $\infty$ $0.5$

A fair six-sided die is rolled twice. Let $X$ be the maximum of the two outcomes. Find the expected value of $X$. $3.25$ $4.25$ $4.47$ $5$

Two random variables $X$ and $Y$ have the following joint distribution:\[\begin{array}{|c|c|c|}\hlinex & Y & P(X, Y) \\\hline1 & 2 & 0.3 \\2 & 3 & 0.4 \\3 & 5 & 0.3 \\\hl...

A random variable $X$ has mean $\mu$ and variance $\sigma^2$.A new variable $Y$ is defined as $Y=a X+b$, where $a$ and $b$ are constants.It is known that $\operatorname{V...

A dataset contains the marks (out of $100$) of students in two subjects:\[\begin{array}{|c|c|c|}\hline\text{Student} & \text{Subject A} & \text{Subject B} \\\hline1 & 60 ...

LET $X$ AND $Y$ BE TWO INDEPENDENT RANDOM VARIABLES, BOTH CHOSEN UNIFORMLY AT RANDOM FROM THE INTERVAL $[0,1]$. WHAT IS THE EXPECTED VALUE OF THE ABSOLUTE DIFFERENCE BETW...

YOU ARE PLAYING A GAME WITH A SINGLE FAIR SIX-SIDED DIE. THE RULES ARE:IF YOU ROLL A $k$ WHERE $k \in\{1,2,3\}$, YOU ADD $k$ TO YOUR SCORE AND YOU MUST ROLL AGAIN. IF YOU...