List of practice Questions

For a given sample the computed values of variance and fourth central moment are \(3\) and \(63\) respectively. Then underlying frequency distribution is classified as:

The distance (in meter) for 7 throws of a shotputter are \(14.5,15.2,16.8,17.1,15.9,16.3,14.7\). Calculate sample mean and sample standard deviation.

Let \(X\) and \(Y\) be two independent identically distributed Bernoulli random variables with \[ P(X=1)=\frac12, \qquad P(X=0)=\frac12. \] If \(Z=XY\), then distribution of \(Z\) is:

A student council of \( 10 \) members has two students from the engineering school, three students from the science school, and five students from the arts school. The university administration picks three students from the council at random. What is the chance that among the three chosen students, two belong to the same school and the third belongs to a different school?

\(X\) is a random variable whose mean is 3 and standard deviation is 2. A new random variable \(Y\) is defined as:
\[ Y = \frac{X-3}{4} \]
The standard deviation of \(Y\) is

Which of the following conditions result(s) in a higher statistical power for comparing means of two samples using t-test?

A box contains one red ball and two blue balls. One ball is drawn at random, its colour is noted and the ball is returned to the box. The process is repeated. In the first two draws, blue ball appears each time. The probability of drawing blue ball again in the third draw is

In statistical analysis of numerical data, which ONE or MORE among the options is/are possible value(s) of correlation coefficient?

A box in a machine shop consists of 5 coated and 10 uncoated cutting inserts which are of otherwise similar characteristics. The inserts got mixed up randomly in the box. An operator has taken 4 of them at once without noticing the differences to mount on a 4-tooth face milling cutter that uses inserts.

The probability of the milling cutter having all uncoated inserts is ______ (rounded off to two decimal places).

The number of soldering defects that occur in a semiconductor device follows a discrete Poisson distribution with a probability mass function
\[ p(x) = \frac{e^{-\lambda}\lambda^x}{x!} \]
The average number of soldering defects per semiconductor device is 3.
The probability that a randomly selected semiconductor device will have at least two soldering defects is ______ (rounded off to two decimal places).

Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all
previous coin tosses. Let 𝐸1 be the event that among the second, fourth, and sixth
coin tosses, there are at least two heads. Let 𝐸2 be the event that among the first,
second, third, and fifth coin tosses, there are equal number of heads and tails.
The conditional probability P(𝐸1| 𝐸2) is equal to ____________. (rounded off to
one decimal place)

The probability density function \(f(x)\) of a real-valued random variable \(X\) is
\[f(x)=\frac{1}{3\sqrt{2\pi}}\exp\!\left(-\frac{x^2}{18}\right),\quad x\in(-\infty,+\infty).\]
Which one of the following statements is correct about the random variable X?

Let \(M\) be a randomly chosen non-empty subset of \(S = \{1,2,3,\ldots,2026\}\).
Which of the following is the probability that the product of all the elements of \(M\) is even?

Suppose that a computer program provides a non-negative and integer-valued random solution to the equation \(n_1 + n_2 + n_3 + n_4 = 20\).
Which of the following is the probability that all of \(n_1, n_2, n_3, n_4\) in the provided solution are positive?

Suppose a random variable \(Z\) follows \(Normal(\mu=0, \sigma^2=1)\) distribution with probability density function \(g(z)\) and cumulative distribution function \(G(z)\). Another random variable \(Y\) follows \(t_1\) distribution with probability density function \(h(y)\) and cumulative distribution function \(H(y)\). Let \(c\) be the positive real number for which \(g(c) = h(c)\).

Which of the following statements is/are correct?

Let \(X\) be an exponentially distributed random variable with mean \(\lambda (> 0)\). If \(P(X > 5) = 0.35\), then the conditional probability \(P(X > 10 \mid X > 5)\) is ________. (Rounded off to two decimal places)

Let \(X\) and \(Y\) be two independent random variables. \(X\) follows \(Bernoulli(p=0.3)\) distribution and \(Y\) follows \(Normal(\mu=0, \sigma^{2}=100)\) distribution.

Which of the following options is the variance of \((2X-1)Y\)?

Let
\[ L = \lim_{n \to \infty} \sum_{k=0}^{n} \frac{e^{-n} n^{k}}{k!} \]
Which of the following is the value of \(L\)?

Let \(X_1, X_2, \ldots, X_n\) be \(n\) independent random variables. Each of the random variables follows \(Normal(\mu=0, \sigma^2=1)\) distribution. Define \(\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i\).

Which of the following statements is/are correct?

Let \(X\) be a discrete valued random variable with cumulative distribution function \(F(x)\).

Which of the following statements is/are correct?

A clinic specializes in testing for a disease \(D\). The result of the test can be either positive or negative.

A study revealed that if a person suffers from the disease \(D\), the test result in that clinic comes out positive 80% of the time, and negative 20% of the time. If a person is not suffering from the disease \(D\), the test comes out positive 10% of the time and negative 90% of the time. It is also known that among the general population, the disease \(D\) occurs in 30% of the individuals.

If a person tests positive for \(D\) in that clinic, the probability that he/she actually suffers from the disease \(D\) is _______. (Rounded off to two decimal places)

For a given data set \(\{x_1, x_2, \ldots, x_n\}\), where \(n = 100\), it is known that
\[ \frac{1}{2000}\sum_{i=1}^{n}\sum_{j=1}^{n}(x_i - x_j)^2 = 99 \]
Let us denote \(\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i\)

The value of \(\frac{1}{99}\sum_{i=1}^{n}(x_i - \bar{x})^2\) is _______. (Answer in integer)

Let \(X\) be a random variable that follows \(Uniform(-1,1)\) distribution. The conditional distribution of the random variable \(Y\) given \(X = x\) is the \(Uniform(x^2-0.1, x^2+0.1)\) distribution.

The value of \(correlation(X,Y)\) is _______. (Answer in integer)

Let \(A_{5\times5}\) be a matrix such that each of its elements follows \(Bernoulli(p=0.50)\) distribution independently.

The probability that the row-sum of the second row and the column-sum of the third column are both equal to 3 is __________. (Rounded off to two decimal places)

Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all previous coin tosses. Let \(E_1\) be the event that among the second, fourth, and sixth coin tosses, there are at least two heads. Let \(E_2\) be the event that among the first, second, third, and fifth coin tosses, there are equal number of heads and tails.

The conditional probability \(P(E_1 \mid E_2)\) is equal to ____________. (rounded off to one decimal place)