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.

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.

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