Find parameter from expectation

5 The random variable \(X\) has probability density function, f, given by $$f ( x ) = \begin{cases} \frac { 1 } { x ^ { 2 } } & a < x < b \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are positive constants.

1. It is given that \(\mathrm { E } ( X ) = \ln 2\). Show that \(b = 2 a\).
2. Show that \(a = \frac { 1 } { 2 }\).
3. Find the median of \(X\).

6 The time, \(X\) hours, taken by a large number of people to complete a challenge is modelled by the probability density function given by $$f ( x ) = \left\{ \begin{array} { c l } \frac { 1 } { x ^ { 2 } } & a \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{array} \right.$$ where \(a\) and \(b\) are constants.

1. State what the constants \(a\) and \(b\) represent in this context.
2. Show that \(a = \frac { b } { b + 1 }\).
   It is given that \(\mathrm { E } ( X ) = \ln 3\).
3. Show that \(b = 2\) and find the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-09_2726_35_97_20}
4. Find the median of \(X\).

4 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { \sqrt { } x } & 0 < x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants. It is given that \(\mathrm { E } ( X ) = 3\).

1. Find the value of \(a\) and show that \(k = \frac { 1 } { 6 }\).
2. Find the median of \(X\).

6 Darts are thrown at random at a circular board. The darts hit the board at distances \(X\) centimetres from the centre, where \(X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 2 } { a ^ { 2 } } x & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant.

1. Verify that f is a probability density function whatever the value of \(a\). It is now given that \(\mathrm { E } ( X ) = 8\).
2. Find the value of \(a\).
3. Find the probability that a dart lands more than 6 cm from the centre of the board.

7 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(k\) are constants.

1. Sketch the graph of \(y = \mathrm { f } ( x )\) and explain in non-technical language what this tells you about \(X\).
2. Given that \(\mathrm { E } ( X ) = 4.5\), find
   1. the value of \(a\),
   2. \(\operatorname { Var } ( X )\).

4. A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } k ( a - x ) ^ { 2 } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.

1. Show that \(k a ^ { 3 } = 3\) Given that \(\mathrm { E } ( X ) = 1.5\)
2. use algebraic integration to show that \(a = 6\)
3. Verify that the median of \(X\) is 1.2 to one decimal place. \includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-15_2255_50_314_34}
   

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8 A continuous random variable \(X\) has probability density function given by the following function, where \(a\) is a constant. \(\mathrm { f } ( x ) = \left\{ \begin{array} { l l } \frac { 2 x } { a ^ { 2 } } & 0 \leqslant x \leqslant a , \\ 0 & \text { otherwise. } \end{array} \right\}\) The expected value of \(X\) is 4 .

1. Show that \(a = 6\). Five independent observations of \(X\) are obtained, and the largest of them is denoted by \(M\).
2. Find the cumulative distribution function of \(M\). \section*{OCR} Oxford Cambridge and RSA