Conditional probability with CDF

5. A call centre records the length of time, \(T\) minutes, its customers wait before being connected to an agent. The random variable \(T\) has a cumulative distribution function given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { l r } 0 & t < 0 \\ 0.3 t - 0.004 t ^ { 3 } & 0 \leqslant t \leqslant 5 \\ 1 & t > 5 \end{array} \right.$$

1. Find the proportion of customers waiting more than 4 minutes to be connected to an agent.
2. Given that a customer waits more than 2 minutes to be connected to an agent, find the probability that the customer waits more than 4 minutes.
3. Show that the upper quartile lies between 2.7 and 2.8 minutes.
4. Find the mean length of time a customer waits to be connected to an agent.

1. The continuous random variable \(Y\) has cumulative distribution function given by

$$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0 \\ \frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k \\ \frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$

1. Find \(\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)\)
2. Find the value of \(k\)
3. Use algebraic calculus to find \(\mathrm { E } ( Y )\)

The continuous random variable \(X\) has distribution function given by $$\text{F}(x) = \begin{cases} 0 & x < 0, \\ 1 - e^{-\frac{x}{4}} & x \geqslant 0. \end{cases}$$ For a random value of \(X\), find the probability that \(2\) lies between \(X\) and \(4X\). [3] Find also the expected value of the width of the interval \((X, 4X)\). [4]

A continuous random variable \(X\) has the cumulative distribution function $$F(x) = 0 \quad x < 2,$$ $$F(x) = k(x - a)^2 \quad 2 \leq x \leq 6,$$ $$F(x) = 1 \quad x \geq 6.$$

1. Find the values of the constants \(a\) and \(k\). [4 marks]
2. Show that the median of \(X\) is \(2(1 + \sqrt{2})\). [4 marks]
3. Given that \(X > 4\), find the probability that \(X > 5\). [6 marks]