17M.1.AHL.TZ2.H_7

pestleMathematicsAIHLPaper 117M· ahl-4-17-poisson-distributionsource ↗

The random variable $X$ has the Poisson distribution ${\text{Po}}(m)$ . Given that ${\text{P}}(X > 0) = \frac{3}{4}$ , find the value of $m$ in the form $\ln a$ where $a$ is an integer.

[3]

The random variable $Y$ has the Poisson distribution ${\text{Po}}(2m)$ . Find ${\text{P}}(Y > 1)$ in the form $\frac{{b - \ln c}}{c}$ where $b$ and $c$ are integers.

[4]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

${\text{P(}}X > 0) = 1 - {\text{P(}}X = 0)$ (M1)

$\Rightarrow 1 - {{\text{e}}^{ - m}} = \frac{3}{4}$ or equivalent A1

$\Rightarrow m = \ln 4$ A1

[3 marks]

${\text{P}}(Y > 1) = 1 - {\text{P}}(Y = 0) - {\text{P}}(Y = 1)$ (M1)

$= 1 - {{\text{e}}^{ - 2\ln 4}} - {{\text{e}}^{ - 2\ln 4}} \times 2\ln 4$ A1

recognition that $2\ln 4 = \ln 16$ (A1)

${\text{P}}(Y > 1) = \frac{{15 - \ln 16}}{{16}}$ A1

[4 marks]

Examiners’ report

[N/A]