CDF to PDF derivation

2 The continuous random variable \(U\) has (cumulative) distribution function given by $$\mathrm { F } ( u ) = \begin{cases} \frac { 1 } { 5 } \mathrm { e } ^ { u } & u < 0
1 - \frac { 4 } { 5 } \mathrm { e } ^ { - \frac { 1 } { 4 } u } & u \geqslant 0 \end{cases}$$

1. Find the upper quartile of \(U\).
2. Find the probability density function of \(U\).

1

1. The time (in milliseconds) taken by my computer to perform a particular task is modelled by the random variable \(T\). The probability that it takes more than \(t\) milliseconds to perform this task is given by the expression \(\mathrm { P } ( T > t ) = \frac { k } { t ^ { 2 } }\) for \(t \geqslant 1\), where \(k\) is a constant.
   1. Write down the cumulative distribution function of \(T\) and hence show that \(k = 1\).
   2. Find the probability density function of \(T\).
   3. Find the mean time for the task.
2. For a different task, the times (in milliseconds) taken by my computer on 10 randomly chosen occasions were as follows. $$\begin{array} { c c c c c c c c c c } 6.4 & 5.9 & 5.0 & 6.2 & 6.8 & 6.0 & 5.2 & 6.5 & 5.7 & 5.3 \end{array}$$ From past experience it is thought that the median time for this task is 5.4 milliseconds. Carry out a test at the \(5 \%\) level of significance to investigate this, stating your hypotheses carefully.

2 The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < - k
\frac { x + k } { 4 k } & - k \leqslant x \leqslant 3 k
1 & x > 3 k \end{array} \right.$$ where \(k\) is a positive constant.

1. Specify fully, in terms of \(k\), the probability density function of \(X\)
2. Write down, in terms of \(k\), the value of \(\mathrm { E } ( X )\)
3. Show that \(\operatorname { Var } ( X ) = \frac { 4 } { 3 } k ^ { 2 }\)
4. Find, in terms of \(k\), the value of \(\mathrm { E } \left( 3 X ^ { 2 } \right)\)
   

6. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by $$F ( x ) = \left\{ \begin{array} { l c } 0 & x < 0
\frac { x ^ { 2 } } { 20 } ( 9 - 2 x ) & 0 \leqslant x \leqslant 2
1 & x > 2 \end{array} \right.$$

1. Verify that the median of \(X\) lies between 1.23 and 1.24
2. Specify fully the probability density function \(\mathrm { f } ( x )\).
3. Find the mode of \(X\).
4. Describe the skewness of this distribution. Justify your answer.

1. A continuous random variable \(X\) has cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 2
\frac { 1 } { 20 } \left( x ^ { 2 } - 4 \right) & 2 \leqslant x \leqslant 4
\frac { 1 } { 5 } ( 2 x - 5 ) & 4 < x \leqslant 5
1 & x > 5 \end{array} \right.$$

1. Calculate \(\mathrm { P } ( X > 4 )\)
2. Find the probability density function of \(X\), specifying it for all values of \(x\).
3. Find the value of \(a\) such that \(\mathrm { P } ( 3 < X < a ) = 0.642\)
4. Find the probability density function of \(X\), specifying it for all values of \(x\).

6. The continuous random variable X has cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 , & x < 1
\frac { 1 } { 27 } \left( - x ^ { 3 } + 6 x ^ { 2 } - 5 \right) , & 1 \leq x \leq 4
1 , & x > 4 \end{array} \right.$$

1. Find the probability density function \(\mathrm { f } ( x )\).
2. Find the mode of \(X\).
3. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
4. Find the mean \(\mu\) of X .
5. Show that \(\mathrm { F } ( \mu ) > 0.5\).
6. Show that the median of \(X\) lies between the mode and the mean.

6. In an experiment some children were asked to estimate the position of the centre of a circle. The random variable \(D\) represents the distance, in centimetres, between the child's estimate and the actual position of the centre of the circle. The cumulative distribution function of \(D\) is given by $$\mathrm { F } ( d ) = \left\{ \begin{array} { c c } 0 & d < 0
\frac { d ^ { 2 } } { 2 } - \frac { d ^ { 4 } } { 16 } & 0 \leqslant d \leqslant 2
1 & d > 2 \end{array} \right.$$

1. Find the median of \(D\).
2. Find the mode of \(D\). Justify your answer. The experiment is conducted on 80 children.
3. Find the expected number of children whose estimate is less than 1 cm from the actual centre of the circle.

8 The continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant - 4
\frac { x + 4 } { 9 } & - 4 \leqslant x \leqslant 5
1 & x \geqslant 5 \end{array} \right.$$

1. Determine the probability density function, \(\mathrm { f } ( x )\), of \(X\).
2. Sketch the graph of f .
3. Determine \(\mathrm { P } ( X > 2 )\).
4. Evaluate the mean and variance of \(X\).

6 The continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0
\frac { 1 } { 2 } x - \frac { 1 } { 16 } x ^ { 2 } & 0 \leqslant x \leqslant 4
1 & x > 4 \end{cases}$$

1. Find the probability that \(X\) lies between 0.4 and 0.8 .
2. Show that the probability density function, \(\mathrm { f } ( x )\), of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } - \frac { 1 } { 8 } x & 0 \leqslant x \leqslant 4
   0 & \text { otherwise } \end{cases}$$
3. 1. Find the value of \(\mathrm { E } ( X )\).
   2. Show that \(\operatorname { Var } ( X ) = \frac { 8 } { 9 }\).
4. The continuous random variable \(Y\) is defined by $$Y = 3 X - 2$$ Find the values of \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).

7 The continuous random variable \(X\) has a cumulative distribution function \(\mathrm { F } ( x )\), where $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1
\frac { 1 } { 4 } ( x - 1 ) & 1 \leqslant x < 4
\frac { 1 } { 16 } \left( 12 x - x ^ { 2 } - 20 \right) & 4 \leqslant x \leqslant 6
1 & x > 6 \end{array} \right.$$

1. Sketch the probability density function, \(\mathrm { f } ( x )\), on the grid below.
2. Find the mean value of \(X\).

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

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