6. The number of A-grades, \(X\), achieved in total by students at Lowkey School in their Mathematical examinations each year can be modelled by a Poisson distribution with a mean of 3 .
a. Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A -grades in their Mathematics examinations.
b. The number of A-grades, \(Y\), achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with mean of 7 . Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A-grades in their Mathematics and English examinations.
c. Lowkey School is given a performance rating, \(P = 2 X + 3 Y\), based on the number of A-grades achieved in Mathematics and English. Find: $$\begin{array} { l l } \text { i. } & \mathrm { E } ( P )
\text { ii. } & \operatorname { Var } ( P ) \end{array}$$ d. What assumption did you make in answering part (b)? Did you need this assumption to answer part (c)? Justify your answers.