The random variable X follows a Poisson distribution with mean m and satisfies

\[{\text{P}}(X = 1) + {\text{P}}(X = 3) = {\text{P}}(X = 0) + {\text{P}}(X = 2).\]

(a)     Find the value of m correct to four decimal places.

(b)     For this value of m, calculate \({\text{P}}(1 \leqslant X \leqslant 2)\).

(a)     \({\text{P}}(X = 1) + {\text{P}}(X = 3) = {\text{P}}(X = 0) + {\text{P}}(X = 2)\)

\(m{{\text{e}}^{ - m}} + \frac{{{m^3}{{\text{e}}^{ - m}}}}{6} = {{\text{e}}^{ - m}} + \frac{{{m^2}{{\text{e}}^{ - m}}}}{2}\)     (M1)(A1)

\({m^3} - 3{m^2} + 6m - 6 = 0\)     (M1)

\(m = 1.5961\) (4 decimal places)     A1

(b)     \(m = 1.5961 \Rightarrow {\text{P}}(1 \leqslant X \leqslant 2) = m{{\text{e}}^{ - m}} + \frac{{{m^2}{{\text{e}}^{ - m}}}}{2} = 0.582\)     (M1)A1

[6 marks]

Most candidates correctly stated the required equation for m. However, many algebraic errors in the simplification of this equation led to incorrect answers. Also, many candidates failed to find the value of m to the required accuracy, with many candidates giving answers correct to 4 sf instead of 4 dp. In part (b) many candidates did not realize that they needed to calculate \({\text{P}}(X = 1) + {\text{P}}(X = 2)\) and many attempts to calculate other combinations of probabilities were seen.