Moderate -0.3 This is a straightforward application of Poisson distribution properties: scaling the parameter for different time periods (part a: 5 years gives λ=15) and adding independent Poisson variables (part b: X+Y with λ=10). All parts require only direct recall of standard results and calculator work, with no problem-solving insight needed. Slightly easier than average due to its routine nature, though the multi-part structure and need to recognize the scaling/addition properties prevents it from being significantly below average.
4. 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 .
i. 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.
ii. What assumption did you make in answering part (b)(i)? [0pt]
[Total 7 marks]
4. 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 .\\
i. 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.\\
ii. What assumption did you make in answering part (b)(i)?\\[0pt]
[Total 7 marks]\\
\hfill \mbox{\textit{SPS SPS FM Statistics 2020 Q4 [7]}}