Expectation from CDF/PDF

20\% of the t -shirts made by the manufacturer are small and sell for \(\pounds 10 30 \%\) of the t -shirts made by the manufacturer are medium and sell for \(\pounds 12\) The rest of the t -shirts made by the manufacturer are large and sell for \(\pounds 15\)

1. Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
2. List all the possible combinations of the individual selling prices of these 3 t-shirts.
3. Find the sampling distribution of the median selling price of these 3 t-shirts.
   1. A supermarket receives complaints at a mean rate of 6 per week.
   2. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
   3. Find the probability that, in a given week, there are
      1. fewer than 3 complaints received by the supermarket,
      2. at least 6 complaints received by the supermarket.
   In a randomly selected week, the supermarket received 12 complaints.
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
   State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
5. Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.
   1. The continuous random variable \(Y\) has cumulative distribution function given by
   $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0 \\ \frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k \\ \frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$
6. Find \(\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)\)
7. Find the value of \(k\)
8. Use algebraic calculus to find \(\mathrm { E } ( Y )\)
   1. The discrete random variable \(X\) is given by
   $$X \sim \mathrm {~B} ( n , p )$$ The value of \(n\) and the value of \(p\) are such that \(X\) can be approximated by a normal random variable \(Y\) where $$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$ Given that when using a normal approximation $$\mathrm { P } ( X < 86 ) = 0.2266 \text { and } \mathrm { P } ( X > 97 ) = 0.1056$$
9. show that \(\sigma = 6\)
10. Hence find the value of \(n\) and the value of \(p\)

2 A continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \frac { 1 } { x ^ { 3 } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ Find \(E ( \sqrt { } X )\).

The length of time a battery works, in tens of hours, is modelled by a random variable \(X\) with cumulative distribution function $$F(x) = \begin{cases} 0 & \text{for } x < 0, \\ \frac{x^3}{432}(8-x) & \text{for } 0 \leq x \leq 6, \\ 1 & \text{for } x > 6. \end{cases}$$

1. Find \(P(X > 5)\). [2]
2. A head torch uses three of these batteries. All three batteries must work for the torch to operate. Find the probability that the head torch will operate for more than 50 hours. [2]
3. Show that the upper quartile of the distribution lies between 4·5 and 4·6. [3]
4. Find \(f(x)\), the probability density function for \(X\). [3]
5. Find the mean lifetime of the batteries in hours. [4]
6. The graph of \(f(x)\) is given below. \includegraphics{figure_1} Give a reason why the model may not be appropriate. [1]