Find multiple parameters from system

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\includegraphics[max width=\textwidth, alt={}, center]{a9f9cf66-0734-4316-99ae-c57090d08135-08_353_1141_255_463} The diagram shows the continuous random variable \(X\) with probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 128 } \left( 4 a x - b x ^ { 3 } \right) & 0 \leqslant x \leqslant 4
c & 4 \leqslant x \leqslant 6
0 & \text { otherwise } \end{cases}$$ where \(a , b\) and \(c\) are constants.
The upper quartile of \(X\) is equal to 4 .

1. Show that \(c = \frac { 1 } { 8 }\) and find the values of \(a\) and \(b\).
2. Find the exact value of the median of \(X\).
3. Find \(\mathrm { E } ( \sqrt { X } )\), giving your answer correct to 2 decimal places.

6 The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} a + b x & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(2 a + 2 b = 1\).
2. It is given that \(\mathrm { E } ( X ) = \frac { 11 } { 9 }\). Use this information to find a second equation connecting \(a\) and \(b\), and hence find the values of \(a\) and \(b\).
3. Determine whether the median of \(X\) is greater than, less than, or equal to \(\mathrm { E } ( X )\).

7 A continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } a x ^ { - 3 } + b x ^ { - 4 } & x \geqslant 1
0 & \text { otherwise } \end{array} \right.$$ where \(a\) and \(b\) are constants.

1. Explain what the letter \(x\) represents. It is given that \(\mathrm { P } ( X > 2 ) = \frac { 3 } { 16 }\).
2. Show that \(a = 1\), and find the value of \(b\).
3. Find \(\mathrm { E } ( X )\).

5. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( x ^ { 2 } + a \right) & - 1 < x \leqslant 2
3 k & 2 < x \leqslant 3
0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.
Given that \(\mathrm { E } ( X ) = \frac { 17 } { 12 }\)

1. find the value of \(k\) and the value of \(a\)
2. Write down the mode of \(X\)

5. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( x ^ { 2 } + a \right) & - 1 < x \leqslant 2
3 k & 2 < x \leqslant 3
0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.
Given that \(\mathrm { E } ( X ) = \frac { 17 } { 12 }\)

1. find the value of \(k\) and the value of \(a\)
2. Write down the mode of \(X\)

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1. The random variable \(Y\) has probability density function \(\mathrm { f } ( y )\) given by

$$\mathrm { f } ( y ) = \left\{ \begin{array} { c c } k y ( a - y ) & 0 \leqslant y \leqslant 3
0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are positive constants.

1. 1. Explain why \(a \geqslant 3\)
   2. Show that \(k = \frac { 2 } { 9 ( a - 2 ) }\) Given that \(\mathrm { E } ( Y ) = 1.75\)
2. show that \(a = 4\) and write down the value of \(k\). For these values of \(a\) and \(k\),
3. sketch the probability density function,
4. write down the mode of \(Y\).

7. The continuous random variable \(X\) has the following probability density function $$f ( x ) = \begin{cases} a + b x & 0 \leqslant x \leqslant 5
0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(10 a + 25 b = 2\) Given that \(\mathrm { E } ( X ) = \frac { 35 } { 12 }\)
2. find a second equation in \(a\) and \(b\),
3. hence find the value of \(a\) and the value of \(b\).
4. Find, to 3 significant figures, the median of \(X\).
5. Comment on the skewness. Give a reason for your answer.

11 The continuous random variable \(X\) has probability density function given by
\(f ( x ) = \begin{cases} a x ^ { 2 } & 0 \leqslant x < 2 ,
b ( 3 - x ) ^ { 2 } & 2 \leqslant x \leqslant 3 ,
0 & \text { otherwise } \end{cases}\)
where \(a\) and \(b\) are positive constants.

1. Given that \(\mathrm { E } ( X ) = 2\), determine the values of \(a\) and \(b\).
2. Determine the median value of \(X\).
3. A random sample of 50 observations of \(X\) is selected. Given that \(\operatorname { Var } ( X ) = 0.2\), determine an estimate of the probability that the mean value of the 50 observations is less than 1.9.