PDF to CDF derivation

1 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } \left( x ^ { - \frac { 1 } { 3 } } - x ^ { - \frac { 2 } { 3 } } \right) & 1 \leqslant x \leqslant 27 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the cumulative distribution function of \(X\).
   The random variable \(Y\) is defined by \(Y = X ^ { \frac { 1 } { 3 } }\).
2. Find the probability density function of \(Y\).
3. Find the exact value of the median of \(Y\).

7 The continuous random variable \(T\) has probability density function given by $$f ( t ) = \begin{cases} 4 t ^ { 3 } & 0 < t \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$

1. Obtain the cumulative distribution function of \(T\).
2. Find the cumulative distribution function of \(H\), where \(H = \frac { 1 } { T ^ { 4 } }\), and hence show that the probability density function of \(H\) is given by \(\mathrm { g } ( h ) = \frac { 1 } { h ^ { 2 } }\) over an interval to be stated.
3. Find \(\mathrm { E } \left( 1 + 2 H ^ { - 1 } \right)\).

2

1. A continuous random variable, \(X\), has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 72 } \left( 8 x - x ^ { 2 } \right) & 2 \leqslant x \leqslant 8 \\ 0 & \text { otherwise } \end{cases}$$
   1. Find \(\mathrm { F } ( x )\), the cumulative distribution function of \(X\).
   2. Sketch \(\mathrm { F } ( x )\).
   3. The median of \(X\) is \(m\). Show that \(m\) satisfies the equation \(m ^ { 3 } - 12 m ^ { 2 } + 148 = 0\). Verify that \(m \approx 4.42\).
2. The random variable in part (a) is thought to model the weights, in kilograms, of lambs at birth. The birth weights, in kilograms, of a random sample of 12 lambs, given in ascending order, are as follows. $$\begin{array} { l l l l l l l l l l l l } 3.16 & 3.62 & 3.80 & 3.90 & 4.02 & 4.72 & 5.14 & 6.36 & 6.50 & 6.58 & 6.68 & 6.78 \end{array}$$ Test at the 5\% level of significance whether a median of 4.42 is consistent with these data.

8 The random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 2 \mathrm { e } ^ { - 2 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the distribution function of \(X\).
2. Find the median value of \(X\). The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\).
3. Find the probability density function of \(Y\).

7 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 4 x ^ { 2 } } + \frac { 1 } { 4 } & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the distribution function of \(X\).
2. Find the exact value of the interquartile range of \(X\).

6 The continuous random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 1 \\ \frac { 1 } { 2 } & 1 \leqslant x \leqslant 3 \\ 0 & x > 3 \end{cases}$$ Find the distribution function of \(X\). The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Find

1. the probability density function of \(Y\),
2. the expected value and variance of \(Y\).

1. The continuous random variable \(X\) has probability density function given by

$$f ( x ) = \left\{ \begin{array} { c c } k ( x + 1 ) ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ k ( 6 - 2 x ) & 1 < x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.

1. Sketch the graph of \(\mathrm { f } ( x )\).
2. Show that the value of \(k\) is \(\frac { 3 } { 20 }\)
3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
4. Find the median of \(X\), giving your answer to 3 significant figures.

7. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \begin{cases} \frac { 1 } { 20 } x ^ { 3 } & 0 \leqslant x \leqslant 2 \\ \frac { 1 } { 10 } ( 6 - x ) & 2 < x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(\mathrm { f } ( x )\) for all values of \(x\).
2. Write down the mode of \(X\).
3. Show that \(\mathrm { P } ( X > 2 ) = 0.8\)
4. Define fully the cumulative distribution function \(\mathrm { F } ( x )\). Given that \(\mathrm { P } ( X < a \mid X > 2 ) = \frac { 5 } { 8 }\)
5. find the value of \(\mathrm { F } ( a )\).
6. Hence, or otherwise, find the value of \(a\). Give your answer to 3 significant figures.

1. The continuous random variable \(X\) has probability density function

$$f ( x ) = \begin{cases} 0.1 x & 0 \leqslant x < 2 \\ k x ( 8 - x ) & 2 \leqslant x < 4 \\ a & 4 \leqslant x < 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants.
It is known that \(\mathrm { P } ( X < 4 ) = \frac { 31 } { 45 }\)

1. Find the exact value of \(k\)
2. 1. Find the exact value of \(a\)
   2. Find the exact value of \(\mathrm { P } ( 0 \leqslant X \leqslant 5.5 )\)
3. Specify fully the cumulative distribution function of \(X\)

7. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } \frac { x ^ { 2 } } { 45 } & 0 \leqslant x \leqslant 3 \\ \frac { 1 } { 5 } & 3 < x < 4 \\ \frac { 1 } { 3 } - \frac { x } { 30 } & 4 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{array} . \right.$$

1. Sketch \(\mathrm { f } ( x )\) for \(0 \leqslant x \leqslant 10\)
2. Find the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
3. Find \(\mathrm { P } ( X \leqslant 8 )\).

2. The length of time, in minutes, that a customer queues in a Post Office is a random variable, \(T\), with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } c \left( 81 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 9 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(c\) is a constant.

1. Show that the value of \(c\) is \(\frac { 1 } { 486 }\)
2. Show that the cumulative distribution function \(\mathrm { F } ( t )\) is given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { c c } 0 & t < 0 \\ \frac { t } { 6 } - \frac { t ^ { 3 } } { 1458 } & 0 \leqslant t \leqslant 9 \\ 1 & t > 9 \end{array} \right.$$
3. Find the probability that a customer will queue for longer than 3 minutes. A customer has been queueing for 3 minutes.
4. Find the probability that this customer will be queueing for at least 7 minutes. Three customers are selected at random.
5. Find the probability that exactly 2 of them had to queue for longer than 3 minutes.

6. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 2 x } { 9 } & 0 \leqslant x \leqslant 1 \\ \frac { 2 } { 9 } & 1 < x < 4 \\ \frac { 2 } { 3 } - \frac { x } { 9 } & 4 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Find \(\mathrm { E } ( X )\).
2. Find the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
3. Find the median of \(X\).
4. Describe the skewness. Give a reason for your answer. \includegraphics[max width=\textwidth, alt={}, center]{caaa0133-5b13-4ca7-8e65-8543327c33fd-12_104_61_2412_1884}

6. The continuous random variable \(X\) has a probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } k ( x - 2 ) & 2 \leqslant x \leqslant 3 \\ k & 3 < x < 5 \\ k ( 6 - x ) & 5 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.

1. Sketch the graph of \(\mathrm { f } ( x )\).
2. Show that the value of \(k\) is \(\frac { 1 } { 3 }\)
3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
4. Hence find the 90th percentile of the distribution.
5. Find \(\mathrm { P } [ \mathrm { E } ( X ) < X < 5.5 ]\)

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7 The continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 5 } ( 2 x + 1 ) & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 15 } ( 4 - x ) ^ { 2 } & 1 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. 1. Show that the cumulative distribution function, \(\mathrm { F } ( x )\), for \(0 \leqslant x \leqslant 1\) is $$\mathrm { F } ( x ) = \frac { 1 } { 5 } x ( x + 1 )$$
   2. Hence write down the value of \(\mathrm { P } ( X \leqslant 1 )\).
   3. Find the value of \(x\) for which \(\mathrm { P } ( X \geqslant x ) = \frac { 17 } { 20 }\).
   4. Find the lower quartile of the distribution.

6 The continuous random variable \(X\) has the probability density function defined by $$f ( x ) = \begin{cases} \frac { 3 } { 8 } \left( x ^ { 2 } + 1 \right) & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 4 } ( 5 - 2 x ) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$

1. The cumulative distribution function of \(X\) is denoted by \(\mathrm { F } ( x )\). Show that, for \(0 \leqslant x \leqslant 1\), $$\mathrm { F } ( x ) = \frac { 1 } { 8 } x \left( x ^ { 2 } + 3 \right)$$
2. Hence, or otherwise, verify that the median value of \(X\) is 1 .
3. Show that the upper quartile, \(q\), satisfies the equation \(q ^ { 2 } - 5 q + 5 = 0\) and hence that \(q = \frac { 1 } { 2 } ( 5 - \sqrt { 5 } )\).
4. Calculate the exact value of \(\mathrm { P } ( q < X < 1.5 )\).

7 A continuous random variable \(X\) has the probability density function defined by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 3 } ( 5 - 2 x ) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Sketch the graph of f on the axes below.
2. 1. Find the cumulative distribution function, F , for \(0 \leqslant x \leqslant 1\).
   2. Hence, or otherwise, find the value of the lower quartile of \(X\).
3. 1. Show that the cumulative distribution function for \(1 \leqslant x \leqslant 2\) is defined by $$\mathrm { F } ( x ) = \frac { 1 } { 3 } \left( 5 x - x ^ { 2 } - 3 \right)$$
   2. Hence, or otherwise, find the value of the upper quartile of \(X\). \includegraphics[max width=\textwidth, alt={}, center]{03c1e107-3377-4b0d-9daf-7f70233c18b5-5_554_1050_1217_424}

A random variable \(X\) has probability density function given by $$\text{f}(x) = \begin{cases} -\frac{2}{9}x + \frac{8}{9} & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$

1. Show that the cumulative distribution function F(x) can be written in the form \(ax^2 + bx + c\), for \(1 \leqslant x \leqslant 4\) where \(a\), \(b\) and \(c\) are constants. [3]
2. Define fully the cumulative distribution function F(x). [2]
3. Show that the upper quartile of \(X\) is 2.5 and find the lower quartile. [6]

Given that the median of \(X\) is 1.88

1. describe the skewness of the distribution. Give a reason for your answer. [2]

The continuous random variable \(X\) has the following probability density function: $$f(x) = \begin{cases} \frac{1}{8}x, & 0 \leq x \leq 2, \\ \frac{1}{12}(6-x), & 2 \leq x \leq 6, \\ 0, & \text{otherwise}. \end{cases}$$

1. Sketch \(f(x)\) for all values of \(x\). [4 marks]
2. State the mode of \(X\). [1 mark]
3. Define fully the cumulative distribution function \(F(x)\) of \(X\). [9 marks]
4. Show that the median of \(X\) is 2.536, correct to 4 significant figures. [4 marks]