Recent questions tagged statistics

Is lambda (λ) parameter used in Poisson Distribution, Exponential Distribution the same. If say λ = 4 customers per hour, then Expected value, E(possion X) = rate = λ = 4...

Q.27 Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear map defined by\[T(x_1, x_2, x_3) = (3x_1 + 5x_2 + x_3,\; x_3,\; 2x_1 + 2x_3).\]Then the rank of $T$ is equal to _...

 

For a given data set $\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}$, where $n=100$, it is known that\[\frac{1}{2000} \sum_{i=1}^{n} \sum_{j=1}^{n}\left(x_{i}-x_{j}\right)^{...

Let $$L=\lim_{n\to\infty}\sum_{k=0}^{n}\frac{e^{-n}n^{k}}{k!}.$$Find the value of $L$. 

Let $A$ be a $5 \times 5$ matrix. Assume each element of $A$ is an independent Bernoulli random variable with parameter $p=0.5$, that is, $$A_{ij} \sim \mathrm{Bernoulli...

Let $X \sim \mathrm{Uniform}(-1,1)$.Given $X=x$, let $Y \mid X=x \sim \mathrm{Uniform}(x^2-0.1,\; x^2+0.1)$.Find the value of the correlation $\mathrm{Corr}(X,Y)$. 

Let $X$ and $Y$ be independent random variables.Suppose $X \sim \mathrm{Bernoulli}(p=0.3)$ and $Y \sim \mathcal{N}(\mu=0,\sigma^2=100)$.Find the value of $\mathrm{Var}((2...

In a certain population, $30\%$ of the individuals have Disease $(D)$ and $70\%$ do not have the disease $(ND)$.Among those who have the disease, $80\%$ test positive and...

Let $X$ be a discrete random variable with CDF $F(x)$.Which of the following statements is/are TRUE?$F(x)$ is always a positive function $F(x)$ has jump discontinuity $F(...

From the image, the question says $:$ Let $Z \sim N(0,1)$ with PDF $g(z)$ and CDF $G(z)$. Let $Y \sim t_{1}$ distribution with PDF $h(y)$ and CDF $H(y)$. $g(x) = h(x)$ fo...

Let $X \sim \mathrm{Exp}(\lambda)$. Given that $P(X>5)=0.35$.Find the value of $P(X>10 \mid X>5)$.

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...