PDF with multiple constants

4 A random variable \(X\) has probability density function f defined by $$f ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } - \frac { 18 } { x ^ { 3 } } & 2 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = \frac { 27 } { 2 }\).
2. Show that \(\mathrm { E } ( X ) = \frac { 27 } { 2 } \ln \frac { 3 } { 2 } - 3\).

5
\includegraphics[max width=\textwidth, alt={}, center]{b054d0a0-01b6-4785-807c-851551b90544-06_382_743_260_699} The diagram shows the probability density function, f , of a random variable \(X\), in terms of the constants \(a\) and \(b\).

1. Find \(b\) in terms of \(a\).
2. Show that \(\mathrm { f } ( x ) = \frac { 2 } { a } - \frac { 2 } { a ^ { 2 } } x\).
3. Given that \(\mathrm { E } ( X ) = 0.5\), find \(a\).

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

1. Show that the value of \(k\) is \(\frac { 32 } { 9 }\).
2. Find \(\mathrm { E } ( X )\).
3. Is the median less than or greater than 1.5? Justify your answer numerically.

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

1. Show that \(| a | \leqslant \frac { 1 } { 2 }\).
2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
3. Construct an unbiased estimator \(T _ { 1 }\) of \(a\) based on one observation \(X _ { 1 }\) of \(X\).
4. A second observation \(X _ { 2 }\) is taken. Show that \(T _ { 2 }\), where \(T _ { 2 } = \frac { 3 } { 8 } \left( X _ { 1 } + X _ { 2 } \right)\), is also an unbiased estimator of a.
5. Given that \(\operatorname { Var } ( X ) = \sigma ^ { 2 }\), determine which of \(T _ { 1 }\) and \(T _ { 2 }\) is the better estimator.

3
\includegraphics[max width=\textwidth, alt={}, center]{c4adc528-ae3f-4ea7-9420-d3e1068a85fe-2_524_796_1105_623} The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} a x & 0 < x \leqslant 1
b ( 2 - x ) ^ { 2 } & 1 < x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants. The graph is shown in the above diagram.

1. Find the values of \(a\) and \(b\).
2. Find the value of \(\mathrm { E } \left( \frac { 1 } { X } \right)\).

8 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { k } { x ^ { n } } & x \geqslant 1
0 & \text { otherwise } \end{cases}$$ where \(n\) and \(k\) are constants and \(n\) is an integer greater than 1 .

1. Find \(k\) in terms of \(n\).
2. 1. When \(n = 4\), find the cumulative distribution function of \(X\).
   2. Hence determine \(\mathrm { P } ( X > 7 \mid X > 5 )\) when \(n = 4\).
3. Determine the values of \(n\) for which \(\operatorname { Var } ( X )\) is not defined.

6. A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} a x - b x ^ { 2 } & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ Given that the mode is 1

1. show that \(a = 2 b\)
2. Find the value of \(a\) and the value of \(b\)
3. Calculate F(1.5)
4. State whether the upper quartile of \(X\) is greater than 1.5, equal to 1.5, or less than 1.5 Give a reason for your answer.
   

3. The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} a x + b & 1 \leqslant x < 4
\frac { 3 } { 2 } - \frac { 1 } { 4 } x & 4 \leqslant x \leqslant 6
0 & \text { otherwise } \end{cases}$$ as shown in Figure 1, where \(a\) and \(b\) are constants. \begin{figure}[h] \end{figure}

1. Show that the median of \(X\) is 4
2. Find the value of \(a\) and the value of \(b\)
3. Specify fully the cumulative distribution function of \(X\)

6 In this question solutions relying entirely on calculator technology are not acceptable.
The continuous random variable \(X\) has the following probability density function $$f ( x ) = \begin{cases} a + b x & - 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(4 a + 4 b = 1\) Given that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 17 } { 5 }\)
2. 1. find an equation in terms of \(a\) only
   2. hence show that \(b = 0.1\)
3. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\)
4. Find the value of \(k\) for which \(\mathrm { P } ( X \geqslant k ) = 0.8\)

4. \begin{figure}[h] \end{figure} A continuous random variable \(X\) has the probability density function \(\mathrm { f } ( x )\) shown in Figure 1 $$\mathrm { f } ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 5
k & 5 < x \leqslant 10.5
0 & \text { otherwise } \end{cases}$$ where \(m\) and \(k\) are constants.

1. 1. Show that \(k = \frac { 1 } { 8 }\)
   2. Find the value of \(m\)
2. Find \(\mathrm { E } ( X )\)
3. Find the interquartile range of \(X\)

1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by

$$\mathrm { f } ( x ) = \begin{cases} a x ^ { 3 } & 0 \leqslant x \leqslant 4
b x + c & 4 < x \leqslant d
0 & \text { otherwise } \end{cases}$$ where \(a\), \(b\), \(c\) and \(d\) are constants such that

• \(b x + c = a x ^ { 3 }\) at \(x = 4\)
• \(b x + c\) is a straight line segment with end coordinates ( \(4,64 a\) ) and ( \(d , 0\) )
  1. State the mode of \(X\)

Given that the mode of \(X\) is equal to the median of \(X\)

use algebraic integration to show that \(a = \frac { 1 } { 128 }\)

Find the value of \(d\)

Hence find the value of \(b\) and the value of \(c\)