List of top Questions asked in Indian Institute Of Technology Joint Admission Test for MSc

Let 𝛼 and 𝛽 be real constants such that\(lim_{x-0^+}\frac{∫^x_0(\frac{αt^2}{1+t^4})dt}{βx-sin\,x}\)=1 .
Then, the value of 𝛼 + 𝛽 equals ________

Let 𝑥₁, 𝑥₂, 𝑥₃ and 𝑥₄ be observed values of a random sample from an 𝑁(𝜃, 𝜎 ² ) distribution, where 𝜃∈ℝ and 𝜎>0 are unknown parameters. Suppose that 𝑥̅=\(\frac{1}{4} ∑^4_{i=1} 𝑥_𝑖 = 3.6 \) and \(\frac{1}{3} ∑^4_{i=1} (𝑥_𝑖-\overline{x} )^2= 20.25\) . For testing the null hypothesis 𝐻₀ ∶ 𝜃=0 against 𝐻₁ ∶𝜃≠0, the 𝑝-value of the likelihood ratio test equals

Let 𝑋₁, 𝑋₂, 𝑋₃ be a random sample from an 𝑁(𝜃, 1) distribution, where 𝜃 ∈ ℝ is an unknown parameter. Then, which one of the following conditional expectations does NOT depend on 𝜃 ?

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

Let 𝑓: ℝ→ ℝ be a function defined by 𝑓(𝑥)=𝑥²−𝑥, 𝑥 ∈ ℝ. Let 𝑔: ℝ→ℝ be a twice differentiable function such that 𝑔(𝑥)=0 has exactly three distinct roots in the open interval (0, 1). Let ℎ(𝑥) = 𝑓(𝑥)𝑔(𝑥), 𝑥 ∈ ℝ, and ℎ′′ be the second order derivative of the function ℎ. If 𝑛 is the number of roots of ℎ′′(𝑥)=0 in (0, 1), then the minimum possible value of 𝑛 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 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 𝑓:ℝ×ℝ→ℝ be a function defined by
\(f(x,y)=\begin{cases} \frac{xy(x+y}{x^2+y^2} & \quad \text{if }(x,y)≠(0,0),\\  0, & \quad if\,(x,y)=(0,0). \end{cases}\)
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 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 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 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 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,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 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 (𝑋, 𝑌) be a random vector having the joint probability density function

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

Let the series 𝑆 and 𝑇 be defined by
\(∑^∞_{n-0}\frac{ 2 ⋅ 5 ⋅ 8 ⋯ (3𝑛 + 2) }{1 ⋅ 5 ⋅ 9 ⋯ (4𝑛 + 1)}\) and \(∑^∞ _{n=1}(1 + \frac{1 }{𝑛} ) ^{−𝑛^2}\)
respectively. Then, which one of the following statements is TRUE?

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 (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 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 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 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
\(𝑆_𝑛 =∑_{k=1}^n\frac{1+k2^k}{4^{k-1}},n=1,2,...\)
Then, lim𝑛_→∞ 𝑆_𝑛 equals (round off to two decimal places)

For real constants 𝛼 and 𝛽, suppose that the system of linear equations
𝑥+2𝑦+3𝑧=6; x+ 𝑦 +𝛼𝑧 = 3; 2𝑦+𝑧=𝛽, 
has infinitely many solutions. Then, the value of 4𝛼 + 3𝛽 equals