List of top Standard Discrete and Continuous Distributions Questions asked in GATE ST

Let \( X \) be a discrete random variable with probability mass function \( p \in \{p_0, p_1\} \), where 

To test \( H_0 : p = p_0 \) against \( H_1 : p = p_1 \), the power of the most powerful test of size 0.05, based on \( X \), equals _________ 

Let \( \{X_n\}_{n \geq 0} \) be a time-homogeneous discrete time Markov chain with state space \( \{0, 1\} \) and transition probability matrix

\[ P = \begin{bmatrix} 0.25 & 0.75 \\ 0.75 & 0.25 \end{bmatrix}. \]

If \( P(X_0 = 0) = P(X_0 = 1) = 0.5 \), then

\[ \sum_{k=1}^{100} E[(X_{2k})^2] \]

equals _________ (round off to 2 decimal places).
Let \( \{X_n\}_{n \geq 0} \) be a time-homogeneous discrete time Markov chain with either finite or countable state space \( S \). Then which one of the following statements is true?