List of top Statistics Questions

Let 𝑋₁,𝑋₂, … be a sequence of independent random variables such that
\(p(x_i=1)=\frac{1}{4}\) and \(p(x_i=21)=\frac{3}{4}\) i=1,2,….
For some real constants 𝑐₁ and 𝑐₂, suppose that

Then, the value of \(\sqrt{3}\) (3𝑐₁+𝑐₂) equals

Let 𝑋₁,𝑋₂, … be a sequence of 𝑖. 𝑖. 𝑑. random variables such that
\(P(X_1=0)=p(X_1=1)=P(X_1=2)=\frac{1}{3}\)

Then, the value of 𝛼₁+2𝛼₂+3𝛼₃+4𝛼₄ equals

Let (𝑋, 𝑌) be a random vector having the joint probability density function

where 𝛼 is a positive constant. Then, 𝑃(𝑋 > 𝑌) equal

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, 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 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 ________

A box contains a certain number of balls out of which 80% are white, 15% are blue and 5% are red. All the balls of the same color are indistinguishable. Among all the white balls, 𝛼% are marked defective, among all the blue balls, 6% are marked defective and among all the red balls, 9% are marked defective. A ball is chosen at random from the box. If the conditional probability that the chosen ball is white, given that it is defective, is 0.4, then 𝛼 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 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=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 𝑋₁~Gamma(1,4), 𝑋₂~Gamma(2, 2) and 𝑋₃~Gamma(3, 4) be three independent random variables. If 𝑌=𝑋₁+2𝑋₂+𝑋₃, then 𝐸((\(\frac{y}{4}\))⁴ ) equals ______

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____.

Let 𝑋₁,𝑋₂,𝑋₅ be a random sample from a 𝐵𝑖𝑛(1, 𝜃) distribution, where 𝜃∈(0,1) is an unknown parameter. For testing the null hypothesis 𝐻₀ ∶ 𝜃≤ 0.5 against 𝐻₁ :𝜃>0.5, consider the two tests 𝑇1 and 𝑇2 defined as:
𝑇₁: Reject 𝐻₀ if, and only if, \(∑^5_{i=1}\) 𝑋_𝑖=5. 
𝑇₂: Reject 𝐻₀ if, and only if, \(∑^5_{i=1}\) X_i≥3. 
Let 𝛽_𝑖 be the probability of making Type-II error, at 𝜃=\(\frac{2}{3}\), when the test 𝑇_𝑖 , 𝑖=1,2 , is used. Then, the value of 𝛽₁+𝛽₂ equals ________(round off to two decimal places)

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)

Suppose from the estimation of a linear regression model \(Y_i=\beta_0+\beta_1X_i+e_i\) the residual sum of squares and the total sum of squares are obtained as 44 and 80, respectively. The value of coefficient of determination is ____________ (round off to two decimal places).

Let $X$ be a random variable with pdf $f(x)=\dfrac{1}{2\sqrt{3}}$ for $-\sqrt{3}<x<\sqrt{3}$, and $0$ otherwise. Then, the variance of $X$ is _____(in integer)}.

The following table provides population information for different age groups in 2010 and 2017. The percentage change in old-age dependency ratio from 2010 to 2017 is ____________ (round off to two decimal places).

Age group	Population in 2010	Population in 2017
0 to 14 years	201630	213609
15 to 64 years	899210	847552
65 years and above	232450	254474

Which of the following is/are used for testing the assumption of normality?

Correlation coefficient lies between ______________ and ________________, and probability lies between ______________ and ________________.

If standard deviation is 0.3, then its corresponding variance would be ______

If the variance of a set of scores is 100, the standard deviation is _______ (in integer).