List of top Data Science and Artificial Intelligence Questions on Probability and Statistics

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)