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]