List of top Statistics Questions

Let 𝑋₁, 𝑋₂, 𝑋₃, 𝑋4 be a random sample of size 4 from an 𝑁(𝜃, 1) distribution, where 𝜃 ∈ ℝ is an unknown parameter. Let 𝑋̅ = \(\frac{1 }{4 }∑^4_{i=1} X_i\) , 𝑔(𝜃) = 𝜃 ² + 2𝜃 and 𝐿(𝜃) be the Cramer-Rao lower bound on variance of unbiased estimators of 𝑔(𝜃). Then, which one of the following statements is FALSE?

Let 𝑋 be a random variable having a probability mass function 𝑝(𝑥) which is positive only for non-negative integers. If
𝑝(𝑥+1)=(\(\frac{ ln\,\, 3}{𝑥+1}\)) 𝑝(𝑥), 𝑥=0, 1, 2, … , 
then 𝑉𝑎𝑟(𝑋) equals

Let 𝑋 be a random variable such that the moment generating function of 𝑋 exists in a neighborhood of zero and
\(E(X^n)=(-1)^n\frac{2}{5}+\frac{2^{n+1}}{5}+\frac{1}{5},\) n=1,2,3,…..
Then, 𝑃 (|𝑋-\(\frac{1}{2}\)|>1) equals

Let 𝑋₁, 𝑋₂,𝑋₁₀ be a random sample from an 𝑁(0, 𝜎² ) distribution, where 𝜎>0 is an unknown parameter. For some real constant 𝑐, let 𝑌=\(\frac{ 𝑐}{10} ∑^{10}_i=1 |𝑋_𝑖 |\) be an unbiased estimator of 𝜎 . Then, the value of 𝑐 equals ________ (round off to two decimal places)

Let 𝑥1=2, 𝑥2=5 and 𝑥3=4 be the observed values of a random sample from a population having a probability mass function
\(f(x;θ) =\begin{cases}  θ(1-θ)^x,   & \quad \text{if }x=0,1,2,...,,\\  0, & \quad Otherwise \end{cases}\)
where 𝜃∈(0, 1) is an unknown parameter. If 𝜏̂ is the uniformly minimum variance unbiased estimate of 𝜃 ² , then 156 𝜏̂ equals ________

Let 𝑋₁, 𝑋₂, 𝑋₃ be 𝑖. 𝑖. 𝑑. random variables, each having the 𝑁(0, 1) distribution. Then, which of the following statements is/are TRUE?

Let 𝑋₁, 𝑋₂ be a random sample from a 𝑈(0, 𝜃) distribution, where 𝜃>0 is an unknown parameter. For testing the null hypothesis 𝐻₀ ∶ 𝜃∈(0,1]∪[2, ∞) against 𝐻₁: 𝜃∈(1, 2), consider the critical region
\(𝑅={(𝑥_1, 𝑥_2 )∈ℝ × ℝ∶\frac{5}{4}<max\,{𝑥_1, 𝑥_2 }<\frac{7}{4}}. \)
Then, the size of the critical region equals____.

A point (𝑎, 𝑏) is chosen at random from the rectangular region [0, 2]×[0, 4]. Then, the probability that the area of the region
𝑅={(𝑥, 𝑦) ∈ ℝ × ℝ ∶ 𝑏𝑥 + 𝑎𝑦 ≤ 𝑎𝑏, 𝑥, 𝑦 ≥ 0} 
will be less than 2, equals

Let 𝑋 and 𝑌 be i.i.d. random variables each having the 𝑁(0, 1) distribution. Let 𝑈=\(\frac{𝑋}{𝑌}\) and 𝑍=|𝑈|. Then, which of the following statements is/are TRUE?

Let (𝑋₁, 𝑋 ) be a random vector having the probability mass function

where 𝑐 is a real constant. Then, which of the following statements is/are TRUE?

Based on 10 data points (𝑥₁, 𝑦₁ ), (𝑥₂, 𝑦₂ ), … , (𝑥₁₀, 𝑦₁₀) on a variable (𝑋, 𝑌), the simple regression lines of 𝑌 on 𝑋 and 𝑋 on 𝑌 are obtained as 2𝑦 − 𝑥 = 8 and 𝑦−𝑥 =−3, respectively. Let 𝑥̅=\(\frac{ 1}{ 10} ∑^{10}_{i=1} 𝑥_𝑖\) and 𝑦̅ = \(\frac{ 1}{ 10} ∑^{10}_{i=1} y_𝑖\). Then, which of the following statements is/are TRUE?

Let X be a random variable having the probability density function
\(f(x)=\begin{cases} ax^2+b, & 0 \le x \le 3\\ 0, & \text{otherwise,} \end{cases}\)
where a and b are real constants, and \(P(X \ge 2)=\frac{2}{3}.\)
Then E(X) equals __________ (round off to 2 decimal places)

Suppose that four persons enter a lift on the ground floor of a building. There are seven floors above the ground floor and each person independently chooses her exit floor as one of these seven floors. If each of them chooses the topmost floor with probability \(\frac{1}{3}\) and each of the remaining floors with an equal probability, then the probability that no two of them exit at the same floor equals

\(\lim\limits_{n \rightarrow \infin}\frac{6}{n+2}\left\{(2+\frac{1}{n})^2+(2+\frac{2}{n})^2+...+(2+\frac{n-1}{2})^2\right\}\) equals

Let (X, Y) be a random vector having bivariate normal distribution with parameters E(X) = 0, Var(X) = 1, E(Y) = −1, Var(Y) = 4 and ρ(X, Y) = \(-\frac{1}{2}\) , where ρ(X, Y) denotes the correlation coefficient between X and Y. Then P(X + Y > 1 | 2X − Y = 1) equals

Suppose that a fair coin is tossed repeatedly and independently. Let X denote the number of tosses required to obtain for the first time a tail that is immediately preceded by a head. Then E(X) and P(X > 4), respectively, are

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 X₁ , X₂ , … , X_n(n ≥ 2) be a random sample from Exp(\(\frac{1}{\theta}\)) distribution, where θ > 0 is unknown. If \(\overline{X}=\frac{1}{n}\sum^n_{i=1}X_i\), then which one of the following statements is NOT true ?

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

Let X₁,X₂ , … , X_n(n ≥ 3) be a random sample from a N(μ,σ²) distribution, where μ ∈ \(\R\) and σ > 0 are both unknown. Then which one of the following is a simple null hypothesis ?

Let X₍₁₎ < X₍₂₎ < ⋯ < X₍₉₎ be the order statistics corresponding to a random sample of size 9 from U(0, 1) distribution. Then which one of the following statements is NOT true ?

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 ?

Let 2.5, -1.0, 0.5, 1.5 be the observed values of a random sample of size 4 from a continuous distribution with the probability density function
\(f(x)=\frac{1}{8}e^{-|x-2|}+\frac{3}{4\sqrt{2\pi}}e^{-\frac{1}{2}(x-\theta)^2},\ \ x \in \R,\)
where θ ∈ \(\R\) is unknown. Then the method of moments estimate of θ equals __________ (round off to 2 decimal places)

Let X and Y be two independent random variables having N(0, \(σ^2_1\)) and N(0, \(σ^2_2\)) distributions, respectively, where 0 < σ₁ < σ₂. Then which of the following statements is/are true ?