5.03f Relate pdf-cdf: medians and percentiles

The continuous random variable \(X\) has the following cumulative distribution function: $$\text{F}(x) = \begin{cases} 0, & x < 0, \\ \frac{1}{432} x^2(x^2 - 16x + 72), & 0 \leq x \leq 6, \\ 1, & x > 6. \end{cases}$$

1. Find P(\(X < 2\)). [2 marks]
2. Find and specify fully the probability density function f(\(x\)) of \(X\). [4 marks]
3. Show that the mode of \(X\) is 2. [6 marks]
4. State, with a reason, whether the median of \(X\) is higher or lower than the mode of \(X\). [1 mark]

The length of time, in tens of minutes, that patients spend waiting at a doctor's surgery is modelled by the continuous random variable \(T\), with the following cumulative distribution function: $$F(t) = \begin{cases} 0, & t < 0, \\ \frac{1}{135}(54t + 9t^2 - 4t^3), & 0 \leq t \leq 3, \\ 1, & t > 3. \end{cases}$$

1. Find the probability that a patient waits for more than 20 minutes. [3]
2. Show that the median waiting time is between 11 and 12 minutes. [3]
3. Define fully the probability density function f(t) of \(T\). [3]
4. Find the modal waiting time in minutes. [4]
5. Give one reason why this model may need to be refined. [1]

The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{4}{45}(x^3 - 10x^2 + 29x - 20) & 1 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases}$$

1. Find P\((X < 2)\) [2 marks]
2. Verify that the median of \(X\) is 2.3, correct to two significant figures. [4 marks]
3. Find the mean of \(X\). [2 marks]

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]

A continuous random variable \(X\) has cumulative distribution function \(F\) given by $$F(x) = \begin{cases} 0 & \text{for } x < 0, \\ \frac{1}{4}x & \text{for } 0 \leqslant x \leqslant 2, \\ \frac{1}{480}x^4 + \frac{7}{15} & \text{for } 2 < x \leqslant b, \\ 1 & \text{for } x > b. \end{cases}$$

1. Show that \(b = 4\). [2]
2. Find P\((X \leqslant 2 \cdot 5)\). [2]
3. Write down the value of the lower quartile of \(X\). [1]
4. Find the value of the upper quartile of \(X\). [3]
5. Find, correct to three significant figures, the value of \(k\) that satisfies the equation P\((X > 3 \cdot 5) = \text{P}(X < k)\). [4]

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]

The continuous random variable \(X\) has cumulative distribution function given by $$F(x) = \begin{cases} 0 & x \leq 0 \\ k\left(x^3 - \frac{3}{8}x^4\right) & 0 < x \leq 2 \\ 1 & x > 2 \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{1}{2}\) [1]
2. Showing your working clearly, use calculus to find
   1. E(\(X\))
   2. the mode of \(X\)
   [6]

The length of time, in years, that a salesman keeps his company car may be modelled by the continuous random variable \(T\). The probability density function of \(T\) is given by $$f(t) = \begin{cases} \frac{2}{5}t e^{-\frac{t^2}{5}} & t \geq 0, \\ 0 & \text{otherwise}. \end{cases}$$

1. Sketch the graph of \(f(t)\). [2]
2. Find the cumulative distribution function \(F(t)\) and hence find the median value of \(T\). [3]
3. Find the probability that \(T\) is greater than the modal value of \(T\). [5]
4. The probability that a randomly chosen salesman keeps his car longer than \(N\) years is \(0.05\). Find the value of \(N\) correct to \(3\) significant figures. [3]