7 The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).
- Prove, from first principles, that \(\mathrm { E } ( X ) = \lambda\).
- Hence, given that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\), find, in terms of \(\lambda\), an expression for \(\operatorname { Var } ( X )\).
- The mode, \(m\), of \(X\) is such that
$$\mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m - 1 ) \quad \text { and } \quad \mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m + 1 )$$
- Show that \(\lambda - 1 \leqslant m \leqslant \lambda\).
- Given that \(\lambda = 4.9\), determine \(\mathrm { P } ( X = m )\).
- The random variable \(Y\) has a Poisson distribution with mode \(d\) and standard deviation 15.5.
Use a distributional approximation to estimate \(\mathrm { P } ( Y \geqslant d )\).
\includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-19_2484_1709_223_153}