List of top Mathematics Questions on Probability and Statistics

The number of accidents per week in a town follows a Poisson distribution with mean 2. If the probability that there are three accidents in two weeks time is \( k e^{-6} \), then the value of \( k \) is:
Which one of the following is NOT a correct statement?
The scores of students in a national level examination are normally distributed with a mean of 500 and a standard deviation of 100. If the value of the cumulative distribution of the standard normal random variable at 0.5 is 0.691, then the probability that a randomly selected student scored between 450 and 500 is:

The probability distribution of the random variable X is given by

X0123
P(X)0.2k2k2k

Find the variance of the random variable \(X\).

What is the mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on 1 face?
What is the mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on 1 face?
The probability distribution of the random variable X is given by
\begin{tabular}{|l|l|l|l|l|} \hline X & 0 & 1 & 2 & 3 \\ \hline P(X) & 0.2 & k & 2k & 2k \\ \hline \end{tabular}
The variance of the random variable X is
The number of accidents occurring in AU region in a month follows Poisson distribution with mean $\lambda = 5$. The probability of less than two accidents in a randomly selected month is ______
Let $X$ be an exponential random variable with mean parameter one. Then the conditional probability $P(X>10 | X>5)$ is equal to
A machine produces 0, 1 or 2 defective items in a day with probabilities of \( \frac{1}{4}, \frac{1}{2}, \frac{1}{4} \) respectively. Then, the standard deviation of the number of defective items produced by the machine in a day is ............
If \( X \) is a continuous random variable with the probability density
\[ f(x) = \begin{cases} K(1 - x^3), & 0 < x < 1 \\ 0, & \text{otherwise} \end{cases} \] Then, the value of \( K \) is ............
The probability of a component being defective is 0.01. There are 100 such components in a machine. Then the probability of two or more defective components in the machine is _______
A number is selected randomly from each of the following two sets:
{1, 2, 3, 4, 5, 6, 7, 8}, {2, 3, 4, 5, 6, 7, 8, 9}
What is the probability that the sum of the numbers is 9?
A machine produces 0, 1 or 2 defective items in a day with probabilities of \( \frac{1}{4}, \frac{1}{2}, \frac{1}{4} \) respectively. Then, the standard deviation of the number of defective items produced by the machine in a day is ...........
Let \( A, B \) be two events and \( \overline{A} \) be the complement of \( A \). If \( P(\overline{A}) = 0.7 \), \( P(B) = 0.7 \), and \( P(B|A) = 0.5 \), then \( P(A \cup B) = \) ...........
If \( A \) and \( B \) are two mutually exclusive events with \( P(B) \ne 1 \), then the conditional probability \[ P(A \mid \overline{B}) = \,? \] where \( \overline{B} \) is the complement of \( B \)
If \( X \) is a continuous random variable with the probability density function
\[ f(x) = \begin{cases} cx^3, & 0 \le x \le 2 \\ 0, & \text{otherwise} \end{cases} \] then \( P\left( \frac{1}{2} < X < \frac{3}{2} \right) \) is ..........
Two numbers \(k_1\) and \(k_2\) are randomly chosen from the set of natural numbers. Then, the probability that the value of \(i^{k_1} + i^{k_2}\) (where \(i = \sqrt{-1}\)) is non-zero equals:
A coin is tossed three times. Let X denote the number of times a tail follows a head. If \(\mu\) and \(\sigma^2\) denote the mean and variance of X, then the value of \(64(\mu + \sigma^2)\) is:
One die has two faces marked 1, two faces marked 2, one face marked 3, and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3, and one face marked 4. The probability of getting the sum of numbers to be 4 or 5 when both the dice are thrown together is:
Let \( f: (0, \infty) \to \mathbb{R} \) be a function which is differentiable at all points of its domain and satisfies the condition \( x^2 f'(x) = 2f(x) + 3 \), with \( f(1) = 4 \). Then \( 2f(2) \) is equal to:
Three defective oranges are accidentally mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \( x \) denotes the number of defective oranges, then the variance of \( x \) is:
Let \( \mathbf{a} = 3\hat{i} - \hat{j} + 2\hat{k} \), \( \mathbf{b} = \mathbf{a} \times (\hat{i} - 2\hat{k}) \) and \( \mathbf{c} = \mathbf{b} \times \hat{k} \). Then the projection of \( \mathbf{c} - 2\hat{j} \) on \( \mathbf{a} \) is:
Let \( A = [a_{ij}] \) be a 2 \(\times\) 2 matrix such that \(a_{ij} \in \{0, 1\}\) for all \(i\) and \(j\). Let the random variable X denote the possible values of the determinant of the matrix A. Then, the variance of X is:
Let \( \mathbf{a} = 3\hat{i} - \hat{j} + 2\hat{k} \), \( \mathbf{b} = \mathbf{a} \times (\hat{i} - 2\hat{k}) \) and \( \mathbf{c} = \mathbf{b} \times \hat{k} \). Then the projection of \( \mathbf{c} - 2\hat{j} \) on \( \mathbf{a} \) is: