Composite/applied transformation

9 A rectangle of area \(A \mathrm {~m} ^ { 2 }\) has a perimeter of 20 m and each of the two shorter sides are of length \(X \mathrm {~m}\), where \(X\) is uniformly distributed between 0 and 2 .

1. Write down an expression for \(A\) in terms of \(X\), and hence show that \(A = 25 - ( X - 5 ) ^ { 2 }\).
2. Write down the probability density function of \(X\).
3. Show that the cumulative distribution function of \(A\) is $$\mathrm { F } ( a ) = \left\{ \begin{array} { l r } 0 & a < 0 ,
   \frac { 1 } { 2 } ( 5 - \sqrt { 25 - a } ) & 0 \leqslant a \leqslant 16 ,
   1 & a > 16 . \end{array} \right.$$
4. Find the probability density function of \(A\). \section*{END OF QUESTION PAPER} \section*{\(\mathrm { OCR } ^ { \text {勾 } }\)}

8 The continuous random variable \(S\) has probability density function given by $$f ( s ) = \begin{cases} \frac { 8 } { 3 s ^ { 3 } } & 1 \leqslant s \leqslant 2
0 & \text { otherwise } \end{cases}$$ An isosceles triangle has equal sides of length \(S\), and the angle between them is \(30 ^ { \circ }\) (see diagram).

1. Find the (cumulative) distribution function of the area \(X\) of the triangle, and hence show that the probability density function of \(X\) is \(\frac { 1 } { 3 x ^ { 2 } }\) over an interval to be stated.
2. Find the median value of \(X\). {www.ocr.org.uk}) after the live examination series.
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Guided tours of a museum begin every 60 minutes. A randomly chosen tourist arrives \(X\) minutes after the start of a tour. The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { ( x - 20 ) ^ { 2 } } { 24000 } & 0 < x < 60
0 & \text { otherwise } \end{cases}$$ The random variable \(T\) is the time that the tourist has to wait for the next tour to begin. Show that the distribution function G of \(T\) is given by $$\mathrm { G } ( t ) = \begin{cases} 0 & t \leqslant 0
\frac { 8 } { 9 } - \frac { ( 40 - t ) ^ { 3 } } { 72000 } & 0 < t < 60
1 & t \geqslant 60 \end{cases}$$ Find the median and the mean of \(T\).

6. The continuous random variable \(X\) has (cumulative) distribution function given by $$F ( x ) = \left\{ \begin{array} { c c } 0 & x < 1
1 - \frac { 1 } { x ^ { 4 } } & x \geq 1 \end{array} \right.$$ a. Show that the probability density function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\), is given by $$g ( y ) = \left\{ \begin{array} { c c } 2 y & 0 < y \leq 1
0 & \text { otherwise } \end{array} \right.$$ b. Find \(\mathrm { E } ( \sqrt [ 3 ] { Y } )\).