List of top Questions asked in IIT JAM MS

A vaccine, when it is administered to an individual, produces no side effects with probability \(\frac{4}{5}\), mild side effects with probability \(\frac{2}{15}\) and severe side effects with probability \(\frac{1}{15}\). Assume that the development of side effects is independent across individuals. The vaccine was administered to 1000 randomly selected individuals. If X₁ denotes the number of individuals who developed mild side effects and X₂ denotes the number of individuals who developed severe side effects, then the coefficient of variation of X₁ + X₂ equals __________ (round off to 2 decimal places)

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables having U(0, 1) distribution. Let Y_n = n min{X₁, X₂ , … , X_n}, n ≥ 1. If Y_n converges to Y in distribution, then the median of Y equals __________ (round off to 2 decimal places)

Let X₍₁₎ < X₍₂₎ < X₍₃₎ < X₍₄₎ < X₍₅₎ be the order statistics based on a random sample of size 5 from a continuous distribution with the probability density function
\(f(x)=\begin{cases} \frac{1}{x^2}, & 1 < x < \infin\\ 0, & \text{otherwise.} \end{cases}\)
Then the sum of all possible values of r ∈ {1, 2, 3, 4, 5} for which E(X_(r)) is finite equals __________

Consider the linear regression model y_i = β₀ + β₁x_i + ∈i , i = 1, 2, … , 6, where β₀ and β₁ are unknown parameters and ∈_i ’s are independent and identically distributed random variables having N(0, 1) distribution. The data on (x_i, y_i) are given in the following table.

xi	1.0	2.0	2.5	3.0	3.5	4.5
yi	2.0	3.0	3.5	4.2	5.0	5.4

If \(\hat{\beta}_0\) and \(\hat{\beta}_1\) are the least squares estimates of β₀ and β₁, respectively, based on the above data, then \(\hat{β}0 + \hat{β}1\) equals __________ (round off to 2 decimal places)

Let X₁, X₂ , … , X₉ be a random sample from a N(μ, σ2) distribution, where μ ∈ \(\R\) and σ > 0 are unknown. Let the observed values of \(\overline{X}=\frac{1}{9}\sum^9_{i=1}X_i\) and \(S^2=\frac{1}{8}\sum^9_{i=1}(X_i-\overline{X})^2\) be 9.8 and 1.44, respectively. If the likelihood ratio test is used to test the hypothesis H₀ : μ = 8.8 against H₁ : μ > 8.8, then the p-value of the test equals __________ (round off to 3 decimal places)