List of practice Questions

Let {a_n}_n≥1 be a sequence of real numbers such that a_1+5m = 2, a_2+5m = 3, a_3+5m = 4, a_4+5m = 5, a_5+5m = 6, m = 0, 1, 2, … . Then limsup_n→∞ a_n + liminf_n→∞ a_n equals __________

Let F : [0, 2] → \(\R\) be the function defined by
\(F(x)=\int_{x^2}^{x+2}e^{x[t]}dt,\)
where [t] denotes the greatest integer less than or equal to t. Then the value of the derivative of F at x = 1 equals

Let f : [1, 2] → \(\R\) be the function defined by
\(f(t)=\int^t_1\sqrt{x^2e^{x^2}-1}\ dx.\)
Then the arc length of the graph of f over the interval [1, 2] equals

Let f : \(\R^2 → \R\) be the function defined by
\(f(x,y) = \begin{cases}   x^2\sin\frac{1}{x}+y^2\cos y, & x \ne 0 \\   0, & x=0. \end{cases}\)
Then which one of the following statements is NOT true ?

Let f : \(\R^2 → \R\) be the function defined by
f(x, y) = 8(x² - y²) - x⁴ + y⁴.
Then which of the following statements is/are true ?

Let f : \(\R → \R\) be a function such that
20(x - y) ≤ f(x) - f(y) ≤ 20(x - y) + 2(x - y)²
for all x, y ∈ \(\R\) and f(0) = 2. Then f(101) equals __________

Let f : \(\R→\R\) be the function defined by
\(f(x)=\begin{cases} \lim\limits_{h \rightarrow0}\frac{(x+h)\sin(\frac{1}{x}+h)-x\sin\frac{1}{x}}{h} & x \ne 0\\ 0, & x=0\end{cases}\)
Then which one of the following statements is NOT true ?

Let f : \(\R → \R\) be the function defined by
\(f(x)=\text{det}\begin{pmatrix}1+x & 9 & 9 \\ 9 & 1+x & 9 \\ 9 & 9 & 1+x \end{pmatrix}\)
Then the maximum value of f on the interval [9, 10] equals

Let the system of equations
x + ay + z = 1
2x + 4y + z = -b
3x + y + 2z = b + 2
have infinitely many solutions, where a and b are real constants. Then the value of 2a + 8b equals

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 __________

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 ?

Let X and Y be two independent and identically distributed random variables having U(0, 1) distribution. Then P(X² < Y < X) 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 ?

Let X be a random variable denoting the amount of loss in a business. The moment generating function of X is
\(M(t)=(\frac{2}{2-t})^2,\ \ \ \ t< 2.\)
If an insurance policy pays 60% of the loss, then the variance of the amount paid by the insurance policy equals __________ (round off to 2 decimal places)

Let X be a random variable having binomial distribution with parameters n(> 1) and p(0 < p < 1). Then \(E(\frac{1}{1+X})\) equals

Let X be a random variable with the moment generating function
\(M(t)=\frac{1}{(1-4t)^5},t \lt\frac{1}{4}.\)
Then the lower bounds for P(X < 40), using Chebyshev’s inequality and Markov’s inequality, respectively, are

Let (X, Y) be a discrete random vector. Then which of the following statements is/are true ?

Let (X, Y) be a random vector having the joint moment generating function
\(M(t_1,t_2)=(\frac{1}{2}e^{-t_1}+\frac{1}{2}e^{t_1})^2(\frac{1}{2}+\frac{1}{2}et_2)^2,\ \ (t_1,t_2)\in \R^2.\)
Then P( |X + Y| = 2) equals __________ (round off to 2 decimal places)

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 and X₂ be two independent and identically distributed random variables having \(X^2_2\) distribution and W = X₁ + X₂ . Then P(W > E(W)) equals __________ (round off to 2 decimal places)

Let X₁,X₂ and X₃ be three independent and identically distributed random variables having U(0, 1) distribution. Then \(E[(\frac{\ln X_1}{\ln X_1 X_2 X_3})^2]\) equals

Let X₁, X₂ and X₃ be three independent and identically distributed random variables having N(0, 1) distribution. If
\(U=\frac{2X^2_1}{(X_2+X_3)^2}\) and \(V=\frac{2(X_2-X_3)^2}{2X^2_1+(X_2+X_3)^2},\)
then which of the following statements is/are true ?

Let X₁ ,X₂ , … , X₁₆ be a random sample from a N(4μ, 1) distribution and Y₁ ,Y₂ , … , Y₈ be a random sample from a N(μ, 1) distribution, where μ ∈ \(\R\) is unknown. Assume that the two random samples are independent. If you are looking for a confidence interval for μ based on the statistic \(8\overline{X} + \overline{Y}\), where \(\overline{X}=\frac{1}{16}\sum^{16}_{i=1}X_i\) and \(\overline{Y}=\frac{1}{8}\sum^8_{i=1}Y_i\), then which one of the following statements is true ?

Let X₁, X₂ ,…, X₂₅ be a random sample from a 𝑁(μ, 1) distribution, where μ ∈ \(\R\) is unknown. Consider testing of the hypothesis H₀ : μ = 5.2 against H₁ : μ = 5.6. The null hypothesis is rejected if and only if \(\frac{1}{25}\sum^{25}_{i=1}X_i > k\) , for some constant k. If the size of the test is 0.05, then the probability of type-II error equals __________ (round off to 2 decimal places)