Sampling - Central Limit Theorem

The probability distribution function of a discrete random variable, \(X\), is given as

\[\mathrm{P}(X = x) = \frac{k}{x!},\text{ for}\hspace{0.5em} x = 0, 1, 2, 3, \dots,\]

where \(k\) is a positive constant.

1. By considering an expansion found in the list of formulae (MF27), show that \(k = \frac{1}{\mathrm{e}}\).[2]
2. Find the value of \(\mathrm{E}(X)\).[3]

It is given that \(\mathrm{Var}(X) = \mathrm{E}(X)\).

3. Find the probability that, in a random sample of 50 observations of \(X\), the mean of the sample is at least 0.95.[3]