Expectation of function of X

3 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 81 } x ^ { 2 } & 0 \leqslant x \leqslant 9 \\ 1 & x > 9 \end{cases}$$

1. Find \(\mathrm { E } ( \sqrt { X } )\).
2. Find \(\operatorname { Var } ( \sqrt { X } )\).
3. The random variable \(Y\) is given by \(Y ^ { 3 } = X\). Find the probability density function of \(Y\).

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

1. Find \(\mathrm { E } ( \sqrt { X } )\).
   The random variable \(Y\) is given by \(Y = X ^ { 2 }\).
2. Find the probability density function of \(Y\).
3. Find the 40th percentile of \(Y\).

7. The weight, \(X \mathrm {~kg}\), of staples in a bin full of paper has probability density function $$f ( x ) = \left\{ \begin{array} { c c } \frac { 9 x - 3 x ^ { 2 } } { 10 } & 0 \leqslant x < 2 \\ 0 & \text { otherwise } \end{array} \right.$$ Use integration to find

1. \(\mathrm { E } ( X )\)
2. \(\operatorname { Var } ( X )\)
3. \(\mathrm { P } ( X > 1.5 )\) Peter raises money by collecting paper and selling it for recycling. A bin full of paper is sold for \(\pounds 50\) but if the weight of the staples exceeds 1.5 kg it sells for \(\pounds 25\)
4. Find the expected amount of money Peter raises per bin full of paper. Peter could remove all the staples before the paper is sold but the time taken to remove the staples means that Peter will have \(20 \%\) fewer bins full of paper to sell.
5. Decide whether or not Peter should remove all the staples before selling the bins full of paper. Give a reason for your answer.
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4. The continuous random variable, \(X\), has the following probability density function $$f ( x ) = \begin{cases} k x & \text { for } 0 \leqslant x < 1 \\ k x ^ { 3 } & \text { for } 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-3_851_935_826_678}

1. Show that \(k = \frac { 4 } { 17 }\).
2. Determine \(\mathrm { E } ( X )\).
3. Calculate \(\mathrm { E } ( 3 X - 1 )\) and \(\operatorname { Var } ( 3 X - 1 )\).

4

1. \(\text { Find } \mathrm { P } ( X > 1 )\) [0pt] [3 marks]
   4
2. 
   [0pt] [3 marks]
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   4
3. Find \(\mathrm { E } \left( 2 X ^ { - 1 } - 3 \right)\)