List of top Statistics Questions asked in GATE ST

A fair die is rolled twice independently. Let \( X \) and \( Y \) denote the outcomes of the first and second roll, respectively. Then \[ E(X + Y \mid (X - Y)^2 = 1) \] The value of \( E(X + Y \mid (X - Y)^2 = 1) \) is _________ (round off to 2 decimal places).

Evaluate the limit: \[ \lim_{n \to \infty} \frac{(2^n + n 2^n \sin^2 \frac{n}{2})}{(2n - n \cos \frac{1}{n})} \] The value of the limit is _________ (round off to 2 decimal places).

Let \[ I = 4 \int_0^{\frac{1}{\sqrt{2}}} \int_0^x \frac{1}{\sqrt{x^2 + y^2}} \, dy \, dx \] Then the value of \( e^{l+\pi} \) is _________ (round off to 2 decimal places).

Let \( X \) be a non-constant positive random variable such that \( E(X) = 9 \). Then which one of the following statements is true?