List of practice Questions

Write chemical reactions for the following conversions: Ethyl bromide to ethyl methyl ether, Ethyl bromide to ethene, Bromobenzene to toluene, Chlorobenzene to biphenyl.

A sum of money at simple interest amounts to Rs. 815 in 3 years and to Rs. 854 in 4 years. The sum is :

Write the name of the process in Fischer Tropsch process in the synthesis of gasoline.

Let a linear model $ Y = \beta_0 + \beta_1 X + \epsilon $ be fitted to the following data, where $ \epsilon $ is normally distributed with mean 0 and unknown variance $ \sigma^2>0 $: \[ \begin{array}{|c|c|c|c|c|c|} \hline x_i & 0 & 1 & 2 & 3 & 4 \\ \hline y_i & 3 & 4 & 5 & 6 & 7 \\ \hline \end{array} \] Let $ \hat{Y}_0 $ denote the ordinary least-square estimator of $ Y $ at $ X = 6 $, and the variance of $ \hat{Y}_0 $ be $ c\sigma^2 $. Then the value of the real constant $ c $ (rounded off to one decimal place) is equal to ________

Two independent random samples, each of size 7, from two populations yield the following values: \[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline \text{Population 1} & 18 & 20 & 16 & 20 & 17 & 18 & 14 \\ \hline \text{Population 2} & 17 & 18 & 14 & 20 & 14 & 13 & 16 \\ \hline \end{array} \] If Mann–Whitney $ U $ test is performed at 5% level of significance to test the null hypothesis $ H_0: $ Distributions of the populations are same, against the alternative hypothesis $ H_1: $ Distributions of the populations are not same, then the value of the test statistic $ U $ (in integer) for the given data, is ________

Consider a Poisson process $ \{ X(t), t \geq 0 \} $. The probability mass function of $ X(t) $ is given by \[ f(t) = \frac{e^{-4t} (4t)^n}{n!}, \quad n = 0, 1, 2, \dots \] If $ C(t_1, t_2) $ is the covariance function of the Poisson process, then the value of \[ C(5, 3) \text{(in integer) is equal to} ________. \]

Let $ X $ be a random variable with the probability mass function $ p_X(x) = \binom{x-1}{3} \left( \frac{3}{4} \right)^{x-1} \left( \frac{1}{4} \right) $, for $ x = 1, 2, 3, \dots $. Then the value of \[ \sum_{n=0}^{\infty} P(n<X \leq n + 3) \quad \text{(rounded off to two decimal places) is equal to} ________. \]

A real-valued source has PDF $f(x)$ as shown.
A 1-bit quantizer maps positive samples to value $\alpha$ and others to $\beta$.
If $\alpha$ and $\beta$minimize the MSE, then $(\alpha^\ - \beta)$ is ____________
(rounded off to two decimal places).

Consider the two-dimensional vector field $\vec{F}(x,y)=x\,\hat{i}+y\,\hat{j}$, where $\hat{i}$ and $\hat{j}$ denote the unit vectors along the $x$-axis and the $y$-axis, respectively. A contour $C$ in the $xy$-plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral \[ \oint_{C} \vec{F}(x,y)\cdot(dx\,\hat{i}+dy\,\hat{j}) \] is ________________.

Let $X=(X_1,X_2,X_3)^{T}\sim N_3(\mu,\Sigma)$ where \[ \mu=\begin{pmatrix}3\\2\\4\end{pmatrix},\qquad \Sigma=\begin{pmatrix} 4 & 2 & 1\\ 2 & 3 & 0\\ 1 & 0 & 2 \end{pmatrix}. \] Find $E[X_1\mid(X_2=4,X_3=7)]$ (in integer).

Let $X\sim \text{Unif}(0,4)$ and, given $X=x$, $Y\mid X=x\sim \text{Unif}\!\left(0,\dfrac{x^{2}}{4}\right)$. Compute $E(Y^{2})$ (rounded off to three decimal places).

Let $Y_1<\cdots<Y_n$ be order statistics from a continuous distribution symmetric about its mean $\mu$. Find the smallest $n$ (in integer) such that $P(Y_1<\mu<Y_n)\ge 0.99$.

If $P(x,y,z)$ is the point nearest to the origin lying on the intersection of the surfaces $z=xy+5$ and $x+y+z=1$, then the distance (in integer) between the origin and $P$ is __________.

Let $X_1,X_2,X_3,X_4$ be a sample of size 4 from Bernoulli($\theta$), $0<\theta<1$. We test $H_0:\theta=\tfrac14$ vs $H_1:\theta>\tfrac14$, rejecting $H_0$ if $X_1+X_2+X_3+X_4>2$. If $\alpha$ is the Type-I error and $\gamma(\theta)$ the power function, then find $16\alpha+7\gamma\!\left(\tfrac12\right)$ (in integer).

Given $\Phi(1.645)=0.95$ and $\Phi(2.33)=0.99$. For a random sample $X_1,\ldots,X_n\sim N(\mu,2^2)$ with $\mu$ unknown, test $H_0:\mu=10$ vs $H_1:\mu=12$ using a rule that rejects $H_0$ when $\overline X$ is large. Find the smallest $n$ (in integer) so that Type I error is $0.05$ and Type II error is at most $0.01$.

Let $0,1,1,2,0$ be five observations of a random variable $X$ which follows a Poisson distribution with parameter $\theta>0$. Let the minimum variance unbiased estimate of $P(X\le 1)$, based on this data, be $\alpha$. Then $5^{4}\alpha$ (in integer) is equal to $\;__________$

While calculating Spearman’s rank correlation coefficient on $n$ paired observations, it is found that $x_i$ are distinct for all $i\ge 2$, with a single tie $x_1=x_2$, and $\sum_{i=1}^{n} d_i^2=19.5$ where $d_i=\text{rank}(x_i)-\text{rank}(y_i)$. Then the minimum possible value of $\,n^3-n\,$ (in integer) is $\;__________$.

In a laboratory experiment, cats choose between foods A and B. If a cat had A, it will prefer A next time with probability $0.7$; if it had B, it will prefer A next time with probability $0.5$. The experiment is repeated under identical conditions. If $40%$ cats had A in the first experiment, then the percentage (rounded off to one decimal place) of cats that will prefer A in the third experiment is __________.

A random sample of size $5$ is taken from the distribution with density \[ f(x;\theta)= \begin{cases} \dfrac{3x^2}{\theta^3}, & 0[6pt] 0, & \text{elsewhere}, \end{cases} \] where $\theta$ is unknown. If the observations are $3,6,4,7,5$, then the maximum likelihood estimate of the $1/8$ quantile of the distribution (rounded off to one decimal place) is __________.

Consider a gamma distribution with pdf \[ f(x;\beta)= \begin{cases} \dfrac{1}{24\beta^{5}}x^{4}e^{-x/\beta}, & x>0,\\ 0, & \text{elsewhere}, \end{cases} \] with $\beta>0$. For $\beta=2$, find the Cramér–Rao lower bound (rounded to one decimal place) for the variance of any unbiased estimator of $\beta^{2}$ based on a random sample of size 8.

Consider the multiple regression model \[ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \epsilon, \] where $ \epsilon $ is normally distributed with mean 0 and variance $ \sigma^2>0 $, and $ \beta_0, \beta_1, \beta_2, \beta_3 $ are unknown parameters. Suppose 52 observations of $ (Y, X_1, X_2, X_3) $ yield sum of squares due to regression as 18.6 and total sum of squares as 79.23. Then, for testing the null hypothesis $ H_0: \beta_1 = \beta_2 = \beta_3 = 0 $ against the alternative hypothesis $ H_1: \beta_i \neq 0 $ for some $ i = 1, 2, 3 $, the value of the test statistic (rounded off to three decimal places), based on one way analysis of variance, is ________

Suppose a random sample of size 3 is taken from a distribution with the probability density function \[ f(x) = \begin{cases} 2x, & 0<x<1, \\ 0, & \text{elsewhere}. \end{cases} \] If $ p $ is the probability that the largest sample observation is at least twice the smallest sample observation, then the value of $ p $ (rounded off to three decimal places) is ________

Let $ X $ be a random variable such that \[ P \left( \frac{a}{2\pi} X \in \mathbb{Z} \right) = 1, \quad a>0, \] where $ \mathbb{Z} $ denotes the set of all integers. If $ \phi_X(t), t \in \mathbb{R} $, denotes the characteristic function of $ X $, then which of the following is/are true?

Which of the following real-valued functions is/are uniformly continuous on $[0, \infty)$?

Consider the following statements:
(I) Let a random variable $X$ have the probability density function \[ f_X(x) = \frac{1}{2} e^{-|x|}, \quad -\infty<x<\infty. \] Then there exist i.i.d. random variables $X_1$ and $X_2$ such that $X$ and $X_1 - X_2$ have the same distribution.
(II) Let a random variable $Y$ have the probability density function \[ f_Y(y) = \begin{cases} \frac{1}{4}, & -2<y<2, \\ 0, & \text{elsewhere.} \end{cases} \] Then there exist i.i.d. random variables $Y_1$ and $Y_2$ such that $Y$ and $Y_1 - Y_2$ have the same distribution.
Then which of the above statements is/are true?