List of top Statistics Questions

Let 𝑋₁~𝑁(2, 1), 𝑋₂~𝑁(−1,4) and 𝑋₃~𝑁(0, 1) be mutually independent random variables. Then, the probability that exactly two of these three random variables are less than 1, 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, … , 𝑋₅ be 𝑖. 𝑖. 𝑑. random variables, each having the 𝐵𝑖n (1, \(\frac{1}{2}\) ) distribution. Let 𝐾=𝑋₁+𝑋₂+ ⋯ + 𝑋₅ and
\(f(x) =\begin{cases}0,  & \quad \text{if }k=0,\\  x_1+x_2+...+x_k, & \quad if\,k=1,2,...,5. \end{cases}\)
Then, 𝐸(𝑈) equals ________

Let 𝑓∶ℝ→ℝ be a function defined by
𝑓(𝑥)=𝑥² sin(𝑥-1) + 𝑥𝑒^(𝑥−1) , 𝑥∈ℝ . 
Then, 
\(lim_{𝑛→∞} 𝑛 (𝑓 (1+\frac{1}{n} )+𝑓 (1+\frac{2}{𝑛} )+ ⋯ +𝑓(1+\frac{10}{𝑛} )−10)\)
equals ________

Let 𝑋~𝐹_6,2 and 𝑌~𝐹_2,6 . If 𝑃(𝑋 ≤ 2) =\( \frac{216}{343}\) and 𝑃 (𝑌≤\(\frac{1}{2}\) )=𝛼, then 686𝛼 equals

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, 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 𝑋₁,𝑋₂, … , 𝑋_𝑛 be a random sample from a \(u(θ +\frac{σ}{\sqrt3},θ+\sqrt3σ)\) distribution, where 𝜃 ∈ ℝ and 𝜎>0 are unknown parameters. Let 𝑋̅ =\(\frac{ 1}{ 𝑛} ∑^n _{i=1}X_i\) and \(𝑆 =\sqrt \frac{1}{n} ∑^n_{i=1}(X_i-\overline{X})^2\) Let \(\^θ\) and \(\^σ\) be the method of moment estimators of 𝜃 and 𝜎 ,respectively. Then, which one of the following statements is FALSE?

Let 𝑋₁, 𝑋₂, be a sequence of 𝑖.𝑖.𝑑. random variables each having 𝑈(0,1) distribution. Let 𝑌 be a random variable having the distribution function 𝐺. Suppose that
\(lim𝑛_{n→∞} \frac{( 𝑋-1+𝑋_2+𝑋_𝑛}{4} ≤ 𝑥)=𝐺(𝑥)\), for all 𝑥 ∈ ℝ . 
Then, 𝑉𝑎𝑟(𝑌) equals

Let 𝑥₁, 𝑥₂, … , 𝑥₁₀ be the observed values of a random sample of size 10 from an 𝑁(𝜃, 𝜎 2 ) distribution, where 𝜃 ∈ ℝ and 𝜎 > 0 are unknown parameters. If
𝑥̅=\(\frac{1}{10}\) \(∑^{10}_{i=1}x_i=0\) and s=\(\sqrt{\frac{1}{9}∑^{10}_{i=1}(x_i-\overline{x})^2=2,}\)
then based on the values of 𝑥̅ and 𝑠 and using Student’s 𝑡-distribution with 9 degrees of freedom, 90% confidence interval(s) for 𝜃 is/are

Let 𝑋 be a continuous random variable such that 𝑃(𝑋 ≥ 0)=1 and 𝑉𝑎𝑟(𝑋)<∞ . Then, 𝐸(𝑋 ²) is

Let 𝑌~𝐹_4,2 . Then, 𝑃(𝑌≤2) equals

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, 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 (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?

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)

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 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 ________

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 𝐸₁, 𝐸₂ and 𝐸₃ be three events such that 𝑃(𝐸₁∩𝐸₂ )=\(\frac{1}{4}\) , 𝑃(𝐸₁ ∩ 𝐸₃ )=𝑃(𝐸₂ ∩ 𝐸₃ )=\(\frac{1}{5}\) and 𝑃(𝐸₁ ∩ 𝐸₂ ∩ 𝐸₃ ) =\(\frac{1}{6}\) . Then, among the events 𝐸₁ 𝐸₂ and 𝐸₃, the probability that at least two events occur, 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 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 ________