Standard applied PDF calculations

7 A factory is supplied with grain at the beginning of each week. The weekly demand, \(X\) thousand tonnes, for grain from this factory is a continuous random variable having the probability density function given by $$f ( x ) = \begin{cases} 2 ( 1 - x ) & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ Find

1. the mean value of \(X\),
2. the variance of \(X\),
3. the quantity of grain in tonnes that the factory should have in stock at the beginning of a week, in order to be \(98 \%\) certain that the demand in that week will be met.

The length of a telephone call made to a company is denoted by the continuous random variable \(T\). It is modelled by the probability density function $$\text{f}(t) = \begin{cases} kt & 0 \leqslant t \leqslant 10 \\ 0 & \text{otherwise} \end{cases}$$

1. Show that the value of \(k\) is \(\frac{1}{50}\). [3]
2. Find P(\(T > 6\)). [2]
3. Calculate an exact value for E(\(T\)) and for Var(\(T\)). [5]
4. Write down the mode of the distribution of \(T\). [1]

It is suggested that the probability density function, f(\(t\)), is not a good model for \(T\).

1. Sketch the graph of a more suitable probability density function for \(T\). [1]

Light bulbs produced in a certain factory have lifetimes, in 100s of hours, whose distribution is modelled by the random variable \(X\) with probability density function $$f(x) = \frac{2x(3-x)}{9}, \quad 0 \leq x \leq 3;$$ $$f(x) = 0 \quad \text{otherwise}.$$

1. Sketch \(f(x)\). [2 marks]
2. Write down the mean lifetime of a bulb. [1 mark]
3. Show that ten times as many bulbs fail before 200 hours as survive beyond 250 hours. [5 marks]
4. Given that a bulb lasts for 200 hours, find the probability that it will then last for at least another 50 hours. [2 marks]
5. State, with a reason, whether you consider that the density function \(f\) is a realistic model for the lifetimes of light bulbs. [1 mark]

The waiting time, in minutes, at a dentist is modelled by the continuous random variable \(T\) with probability density function $$f(t) = k(10 - t) \quad 0 \leq t \leq 10,$$ $$f(t) = 0 \quad \text{otherwise}.$$

1. Sketch the graph of \(f(t)\) and find the value of \(k\). [4 marks]
2. Find the mean value of \(T\). [4 marks]
3. Find the 95th percentile of \(T\). [3 marks]
4. State whether you consider this function to be a sensible model for \(T\) and suggest how it could be modified to provide a better model. [2 marks]

The queueing times, \(T\) minutes, of customers at a local Post Office are modelled by the probability density function $$f(t) = \frac{1}{2500}t(100-t^2) \quad \text{for } 0 \leq t \leq 10,$$ $$f(t) = 0 \quad \text{otherwise.}$$

1. Determine the mean queueing time. [3]
2. 1. Find the cumulative distribution function, \(F(t)\), of \(T\).
   2. Find the probability that a randomly chosen customer queues for more than 5 minutes.
   3. Find the median queueing time. [10]