Discrete Probability Distributions - Standard Univariate Distributions - Statistics Previous Year Questions

Let \( X \) be a random variable having Poisson distribution such that \( E(X^2) = 110 \). Then which one of the following statements is NOT true?

Let \(X\) be a random variable with the probability mass function \(p_X(x) = \left(\frac{3}{4}\right)^{x-1} \left(\frac{1}{4}\right)\), \(x = 1, 2, 3, ...\). Then the value of
\(\sum_{n=0}^{\infty} P(n < X \leq n+3)\)
(rounded off to two decimal places) is equal to ______

At a telephone exchange, telephone calls arrive independently at an average rate of 1 call per minute, and the number of telephone calls follows a Poisson distribution. Five time intervals, each of duration 2 minutes, are chosen at random. Let \(p\) denote the probability that in each of the five time intervals at most 1 call arrives at the telephone exchange. Then \(e^{10} p\) (in integer) is equal to ______

Let \(X\) be a random variable having Poisson distribution with mean \(\lambda > 0\) such that \(P(X = 4) = 2 P(X = 5)\). If \(p_k = P(X = k)\), \(k = 0, 1, 2, \ldots\), and \(p_{\alpha} = \max_k p_k\), then \(\alpha\) equals __________ (in integer).

Let \( X \sim \text{Bin}\left(2, \frac{1}{5}\right) \). Then \( 18 \, E(X^2) \) equals ______________ (answer in integer).

Let \( X \) be a random variable with support \( S = \{0, 1, 2, ...\} \) and
\( P(X \geq k + 1 | X \geq k) = p, \; k \in S, \; 0 < p < 1. \)
Then which of the following statements is correct?