The random variable X follows a Poisson distribution with mean \(\lambda \).

(a)     Find \(\lambda \) if \({\text{P}}(X = 0) + {\text{P}}(X = 1) = 0.123\).

(b)     With this value of \(\lambda \), find \({\text{P}}(0 < X < 9)\).

(a)     required to solve \({{\text{e}}^{ - \lambda }} + \lambda {{\text{e}}^{ - \lambda }} = 0.123\)     M1A1

solving to obtain \(\lambda = 3.63\)     A2     N2

Note: Award A2 if an additional negative solution is seen but A0 if only a negative solution is seen.

(b)     \({\text{P}}(0 < X < 9)\)

\( = {\text{P}}(X \leqslant 8) - {\text{P}}(X = 0)\) (or equivalent)     (M1)

\( = 0.961\)     A1

[6 marks]

Part (a) - Well done by most, although there were some answers that ignored the requirement of mathematical notation.

Part (b) - Not successfully answered by many. The main problem was not correctly interpreting the inequalities in the probability.