2 The number of televisions of a particular model sold per week at a retail store can be modelled by a random variable \(X\) with the probability function shown in the table.

1. (A) Explain why \(\mathrm { E } ( X ) = 2\).
   (B) Find \(\operatorname { Var } ( X )\).
2. The profit, measured in pounds made in a week, on the sales of this model of television is given by \(Y\), where \(Y = 250 X - 80\).
   Find
   • \(\mathrm { E } ( Y )\) and
   • \(\operatorname { Var } ( Y )\).
   The remote controls for the televisions are quality tested by the manufacturer to see how long they last before they fail.
3. Explain why it would be inappropriate to test all the remote controls in this way.
4. State an advantage of using random sampling in this context. A website awards a random number of loyalty points each time a shopper buys from it. The shopper gets a whole number of points between 0 and 10 (inclusive). Each possibility is equally likely, each time the shopper buys from the website. Awards of points are independent of each other.
5. Let \(X\) be the number of points gained after shopping once. Find
   • the mean of \(X\)
   • the variance of \(X\).
   • Let \(Y\) be the number of points gained after shopping twice.
   Find
   • the mean of \(Y\)
   • the variance of \(Y\).
   • Find the probability of the most likely number of points gained after shopping twice. Justify your answer.
   • State the conditions under which the Poisson distribution is an appropriate model for the number of emails received by one person in a day.
   Jane records the number of junk emails which she receives each day. During working hours (9am to 5pm, Monday to Friday) the mean number of junk emails is 7.4 per day. Outside working hours ( 5 pm to 9am), the mean number of junk emails is 0.3 per hour. For the remainder of this question, you should assume that Poisson models are appropriate for the number of junk emails received during each of "working hours" and "outside working hours".
6. Find the probability that the number of junk emails which she receives between 9am and 5pm on a Monday is
   (A) exactly 10 ,
   (B) at least 10 .
7. (A) What assumption must you make to calculate the probability that the number of junk emails which she receives from 9am Monday to 9am Tuesday is at most 20?
   (B) Find the probability.