List of top Statistics Questions on Univariate Distributions

Let \(X\) be a random variable with the probability density function

Let \(X\) be a random variable with the probability density function

Let \(X\) be a random variable having the moment generating function

Let \(X\) be a positive continuous random variable. Consider the transformation \(Y=X^4\). Then, the Jacobian of the inverse transformation is

Let \( \Theta \) be a random variable having \( U(0, 2\pi) \) distribution. Let \( X = \cos \Theta \) and \( Y = \sin \Theta \). Let \( \rho \) be the correlation coefficient between \( X \) and \( Y \). Then \( 100\rho \) is equal to __________ (answer in integer).

Let \( X_1, X_2, X_3 \) be i.i.d. random variables from a continuous distribution having probability density function
\[f(x) = \begin{cases} \frac{1}{2}x^{-3}, & \text{if } x > \frac{1}{2} \\0, & \text{if } x \leq \frac{1}{2} \end{cases}.\]
Let \( X_{(1)} = \min\{X_1, X_2, X_3\} \). Then the value of \( 10E(X_{(1)}) \) is equal to __________ (answer in integer).

For \( n \in \mathbb{N} \), let \( X_1, X_2, \ldots, X_n \) be a random sample from the Cauchy distribution having probability density function
\[f(x) = \frac{1}{\pi(1 + x^2)}, \quad -\infty < x < \infty.\]
Let \( g: \mathbb{R} \to \mathbb{R} \) be defined by
\[g(x) = \begin{cases} x, & \text{if } -1000 \leq x \leq 1000 \\0, & \text{otherwise}\end{cases}.\]
Let
\[\alpha = \lim_{n \to \infty} P\left( \frac{1}{n^{3/4}} \sum_{i=1}^n g(X_i) > \frac{1}{2} \right).\]
Then \( 100\alpha \) is equal to __________ (answer in integer).

Suppose that the random variable \( X \) has an \( \text{Exp}\left(\frac{1}{5}\right) \) distribution, and for any \( x > 0 \), the conditional distribution of the random variable \( Y \), given \( X = x \), is \( N(x, 2) \). Then \( \text{Var}(X + Y) \) is equal to:

Let \( X \) be a random variable such that \( X \) and \( -X \) have the same distribution. Let \( Y = X^2 \) be a continuous random variable with the probability density function
\[g(y) = \begin{cases} \frac{e^{-y/2}}{\sqrt{2\pi y}}, & \text{if } y > 0 \\0, & \text{if } y \leq 0 \end{cases}.\]
Then \( E((X - 1)^4) \) is equal to:

Let \( f_1(x) \) be the probability density function of the \( N(0,1) \) distribution, and \( f_2(x) \) be the probability density function of the \( N(0, 6) \) distribution. Let \( Y \) be a random variable with probability density function
\[f(x) = 0.6 f_1(x) + 0.4 f_2(x), \quad -\infty < x < \infty.\]
Then \( \text{Var}(Y) \) is equal to:

Let 𝑋 be a continuous random variable having the 𝑈(−2, 3) distribution. Then which of the following statements is correct?

Let (𝑋, 𝑌, 𝑍) be a random vector having the joint probability density function
\(f(x,y, z) =\begin{cases} \frac{1}{2\,xy}, & \quad \text{if }0<z<y<x<1.\\ \frac{1}{2x^2}, & \quad if\,0<z<x<y<2x<2,\\0, & \quad\,\,Otherwise.\end{cases}\)
Then, which one of the following statements is FALSE?

Let 𝑋 be a random variable having a probability density function
\(f(x; θ) =\begin{cases}  (3-θ){x^2-θ}      & \quad \text{if }0<x<1,\\  0, & \quad Otherwise \end{cases}\)
where 𝜃 ∈ {0, 1}. For testing the null hypothesis 𝐻₀ : 𝜃=0 against 𝐻₁ : 𝜃=1, the power of the most powerful test, at the level of significance 𝛼=0.125, equals

Let 𝑋1 and 𝑋2 be 𝑖. 𝑖. 𝑑. random variables having the common probability density function
\(f(x) =\begin{cases}   e^{-x}     & \quad x ≥0,\\  0, & \quad Otherwise \end{cases}\)
Define 𝑋₍₁₎=min{𝑋₁, 𝑋₂} and 𝑋₍₂₎ = max{𝑋₁, 𝑋₂}. Then, which one of the following statements is FALSE?

Let 𝑋 be a random variable having the probability density function
\(f(x) =\begin{cases} \frac{5}{x^6}, & \quad \text{if }x>1,\\ 0, & \quad Otherwise. \end{cases}\)
Then, which of the following statements is/are TRUE for the distribution of 𝑋 ?

Let 𝑋 be a random variable having the probability density function
\(f(x) =\begin{cases}   \frac{x}{2}       & \quad \text{if }0<x<2,\\  0, & \quad Otherwise \end{cases}\)
Then, 𝑉𝑎𝑟 (ln ( \(\frac{2}{x}\) )) equals ________

Let 𝑋₁,𝑋₂, 𝑋₃ be 𝑖.𝑖.𝑑. random variables, each having the 𝑁(2, 4) distribution. If 𝑃(2𝑋₁-3𝑋₂+6𝑋₃>17)=1-Φ(𝛽), then 𝛽 equals ________

Let the probability mass function of a random variable 𝑋 be given by
𝑃(𝑋=𝑛) = \(\frac{𝑘}{(𝑛-1)𝑛}\) , 𝑛=2, 3, … ,
where 𝑘 is a positive constant. Then, 𝑃(𝑋≥17 |𝑋≥5) equals ________

Let the random vector (𝑋₁,𝑋₂) follow the bivariate normal distribution with 𝐸(𝑋₁)=𝐸(𝑋₂ )=1, 𝑉𝑎𝑟(𝑋₂ )=4 𝑉𝑎𝑟(𝑋₁)=4 and 𝐶𝑜𝑣(𝑋₁, 𝑋₂ )=1 . Then, 𝑉𝑎𝑟 (𝑋₁+𝑋₂ | 𝑋₁=\(\frac{1}{2}\)) equals ________

Let 𝑥₁=2.1, 𝑥₂=4.2, 𝑥₃=5.8 and 𝑥₄=3.9 be the observed values of a random sample 𝑋₁, 𝑋₂, 𝑋₃ and 𝑋₄ from a population having a probability density function
\(f(x) =\begin{cases}   \frac{x}{θ^2}e^-\frac{x}{θ}     & \quad \text{if }x >0,\\  0, & \quad Otherwise \end{cases}\)
where 𝜃 ∈ (0, ∞) is an unknown parameter. Then, the maximum likelihood estimate of 𝑉𝑎𝑟(𝑋₁) equals ________

Let 𝑋1, 𝑋2 be a random sample from a distribution having a probability density function
\(f(x) =\begin{cases}   \frac{1}{θ}e^{\frac{y}{θ}}   & \quad \text{if }x >0,\\  0, & \quad Otherwise \end{cases}\)\(𝜃\)
where 𝜃∈(0, ∞) is an unknown parameter. For testing the null hypothesis 𝐻₀ : 𝜃=1 against 𝐻1∶ 𝜃≠1, consider a test that rejects 𝐻₀ for small observed values of the statistic \(𝑊 = \frac{𝑋_1+𝑋_2}{ 2}\) . If the observed values of 𝑋₁ and 𝑋₂ are 0.25 and 0.75, respectively, then the 𝑝-value equals___(round off to two decimal places)

Let (X, Y) be a random vector having the joint probability density function
\(f(x,y)=\begin{cases} \frac{\sqrt2}{\sqrt{\pi}}e^{-2x}e^{-\frac{(y-x)^2}{2}}, & 0 <x< ∞,-∞<y<∞\\ 0,& \text{otherwise} \end{cases}\)
Then E(Y) equals

Let X₁ and X₂ be two independent and identically distributed discrete random variables having the probability mass function
\(f(x) = \begin{cases} (\frac{1}{2})^x, & x=1,2,3,... \\   0, & \text{otherwise. }  \end{cases}\)
Then P(min{X₁ , X₂} ≥ 5) equals

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables having the common probability density function
\(f(x) = \begin{cases} \frac{2}{x^3}, &  x \geq 1 \\     0, & \text{otherwise} . \end{cases}\)
If \(\lim\limits_{n \rightarrow \infin}P(|\frac{1}{n}\sum^n_{i=1}X_i-\theta|< \in)=1\) for all ∈ > 0, then θ equals

A university bears the yearly medical expenses of each of its employees up to a maximum of Rs. 1000. If the yearly medical expenses of an employee exceed Rs. 1000, then the employee gets the excess amount from an insurance policy up to a maximum of Rs. 500. If the yearly medical expenses of a randomly selected employee has U(250, 1750) distribution and 𝑌 denotes the amount the employee gets from the insurance policy, then which of the following statements is/are true ?