Poisson Distribution (CIE A Level Maths: Probability & Statistics 2)

A random variable .

Write down the mean number of occurrences, .

Use calculations of the form to find exact values, in terms of , for

The random variable .

Find the value of  in the case when:

Find the value of  in the case when:

Given that , find a simplified expression in terms of  for:

For each of the following random variables, calculate the mean using  and the standard deviation using .

The random variable .

Factorise the expression

Hence calculate 

Calculate:

A random variable .

Find the exact value, in terms of , of:

Use your answers to part (a) to find the exact value of:

Compare your answers to part (a) and part (b), then use the pattern to write down an expression, in terms of , for

where  is an integer and .

Naomi, a teacher, receives emails at a constant average rate during the school day. Naomi models the number of emails she receives within a fixed period of time using a Poisson distribution, .

Write down two assumptions that Naomi has made about the emails in order to use a Poisson model.

Naomi knows that on average she receives 10 emails in one hour.

Calculate the value of  in the case when Naomi is modelling the number of emails she receives in:

If  and  follow Poisson distributions and are independent then  also follows a Poisson distribution.

In particular, if and  are two independent distributions, then .

The random variables,  are independent.

Write down the distribution of: