Multiple independent observations

6 The length, \(X\) centimetres, of worms of a certain type is modelled by the probability density function $$f ( x ) = \begin{cases} \frac { 6 } { 125 } ( 10 - x ) ( x - 5 ) & 5 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$

1. State the value of \(\mathrm { E } ( X )\).
2. Find \(\operatorname { Var } ( X )\).
3. Two worms of this type are chosen at random. Find the probability that exactly one of them has length less than 6 cm .

6 Alethia models the length of time, in minutes, by which her train is late on any day by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 8000 } ( x - 20 ) ^ { 2 } & 0 \leqslant x \leqslant 20 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the probability that the train is more than 10 minutes late on each of two randomly chosen days.
2. Find \(\mathrm { E } ( X )\).
3. The median of \(X\) is denoted by \(m\). Show that \(m\) satisfies the equation \(( m - 20 ) ^ { 3 } = - 4000\), and hence find \(m\) correct to 3 significant figures.
4. State one way in which Alethia's model may be unrealistic.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The continuous random variable \(X\) has probability density function \(f ( x ) = \begin{cases} k x ^ { n } & 0 \leqslant x \leqslant 1 , \\ 0 & \text { otherwise, } \end{cases}\) where \(k\) is a constant and \(n\) is a parameter whose value is positive. It is given that the median of \(X\) is 0.8816 correct to 4 decimal places. Ten independent observations of \(X\) are obtained. Find the expected number of observations that are less than 0.8 .

3. The lifetime, \(X\), in tens of hours, of a battery is modelled by the probability density function $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 9 } x ( 4 - x ) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$ Use algebraic integration to find

1. \(\mathrm { E } ( X )\)
2. \(\mathrm { P } ( X > 2.5 )\) A radio runs using 2 of these batteries, both of which must be working. Two fully-charged batteries are put into the radio.
3. Find the probability that the radio will be working after 25 hours of use. Given that the radio is working after 16 hours of use,
4. find the probability that the radio will be working after being used for another 9 hours.

1. The lifetime, in tens of hours, of a certain delicate electrical component can be modelled by the random variable \(X\) with probability density function

$$f ( x ) = \begin{cases} \frac { 1 } { 42 } x , & 0 \leq x < 6 \\ \frac { 1 } { 7 } & 6 \leq x \leq 10 \\ 0 , & \text { otherwise } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
2. Find the probability that a component lasts at least 50 hours. A particular device requires two of these components and it will not operate if one or more of the components fail. The device has just been fitted with two new components and the lifetimes of these two components are independent.
3. Find the probability that the device breaks down within the next 50 hours.

The queuing time in minutes, \(X\), of a customer at a post office is modelled by the probability density function $$\text{f}(x) = \begin{cases} kx(81 - x^2) & 0 \leqslant x \leqslant 9 \\ 0 & \text{otherwise} \end{cases}$$

1. Show that \(k = \frac{4}{6561}\). [3]

Using integration, find

1. the mean queuing time of a customer, [4]
2. the probability that a customer will queue for more than 5 minutes. [3]

Three independent customers shop at the post office.

1. Find the probability that at least 2 of the customers queue for more than 5 minutes. [3]

A corner-shop has weekly sales (in thousands of pounds), which can be modelled by the continuous random variable \(X\) with probability density function $$f(x) = k(x-2)(10-x) \quad 2 \leq x \leq 10,$$ $$f(x) = 0 \quad \text{otherwise}.$$

1. Show that \(k = \frac{3}{256}\) and write down the mean of \(X\). [6 marks]
2. Find the standard deviation of the weekly sales. [6 marks]
3. Find the probability that the sales exceed £8 000 in any particular week. [4 marks]

If the sales exceed £8 000 per week for 4 consecutive weeks, the manager gets a bonus.

1. Find the probability that the manager gets a bonus in February. [2 marks]