Multi-part piecewise CDF

5 The continuous random variable \(X\) has cumulative distribution function given by $$F ( x ) = \begin{cases} 0 & x < 1 , \\ \frac { 1 } { 8 } ( x - 1 ) ^ { 2 } & 1 \leqslant x < 3 , \\ a ( x - 2 ) & 3 \leqslant x < 4 , \\ 1 & x \geqslant 4 , \end{cases}$$ where \(a\) is a positive constant.

1. Find the value of \(a\).
2. Verify that \(C _ { X } ( 8 )\), the 8th percentile of \(X\), is 1.8 .
3. Find the cumulative distribution function of \(Y\), where \(Y = \sqrt { X - 1 }\).
4. Find \(C _ { Y } ( 8 )\) and verify that \(C _ { Y } ( 8 ) = \sqrt { C _ { X } ( 8 ) - 1 }\).

1. A continuous random variable \(Y\) has cumulative distribution function given by

$$\mathrm { F } ( y ) = \left\{ \begin{array} { c r } 0 & y < 3 \\ \frac { 1 } { 16 } \left( y ^ { 2 } - 6 y + a \right) & 3 \leqslant y \leqslant 5 \\ \frac { 1 } { 12 } ( y + b ) & 5 < y \leqslant 9 \\ \frac { 1 } { 12 } \left( 100 y - 5 y ^ { 2 } + c \right) & 9 < y \leqslant 10 \\ 1 & y > 10 \end{array} \right.$$ where \(a\), \(b\) and \(c\) are constants.

1. Find the value of \(a\) and the value of \(c\)
2. Find the value of \(b\)
3. Find \(\mathrm { P } ( 6 < Y \leqslant 9 )\) Show your working clearly.
4. Specify the probability density function, f(y), for \(5 < y \leqslant 9\) Using the information $$\int _ { 3 } ^ { 5 } ( 6 y - 5 ) f ( y ) d y + \int _ { 9 } ^ { 10 } ( 6 y - 5 ) f ( y ) d y = 26.5$$
5. find \(\mathrm { E } ( 6 Y - 5 )\) You should make your method clear.

2 The continuous random variable \(H\) has cumulative distribution function given by $$\mathrm { F } ( h ) = \left\{ \begin{array} { l r } 0 & h \leqslant 0 \\ \frac { h ^ { 2 } } { 48 } & 0 < h \leqslant 4 \\ \frac { h } { 6 } - \frac { 1 } { 3 } & 4 < h \leqslant 5 \\ \frac { 3 } { 10 } h - \frac { h ^ { 2 } } { 75 } - \frac { 2 } { 3 } & 5 < h \leqslant d \\ 1 & h > d \end{array} \right.$$ where \(d\) is a constant.

1. Show that \(2 d ^ { 2 } - 45 d + 250 = 0\)
2. Find \(\mathrm { P } ( H < 1.5 \mid 1 < H < 4.5 )\)
3. Find the probability density function \(\mathrm { f } ( h )\) You may leave the limits of \(h\) in terms of \(d\) where necessary.

5. The continuous random variable \(Y\) has cumulative distribution function \(\mathrm { F } ( y )\) given by $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 3 \\ k \left( y ^ { 2 } - 2 y - 3 \right) & 3 \leqslant y \leqslant \alpha \\ 4 k ( 2 y - 7 ) & \alpha < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$ where \(k\) and \(\alpha\) are constants.

1. Find \(\mathrm { P } ( 4.5 < Y \leqslant 5.5 )\)
2. Find the probability density function \(\mathrm { f } ( \mathrm { y } )\)

3. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 0 \\ 4 a x ^ { 2 } & 0 \leqslant x \leqslant 1 \\ a \left( b x ^ { 3 } - x ^ { 4 } + 1 \right) & 1 < x \leqslant 3 \\ 1 & x > 3 \end{array} \right.$$ where \(a\) and \(b\) are positive constants.

1. Show that \(b = 4\)
2. Find the exact value of \(a\)
3. Find \(\mathrm { P } ( X > 2.25 )\)
4. Showing your working clearly,
   1. sketch the probability density function of \(X\)
   2. calculate the mode of \(X\)

1. The continuous random variable \(X\) has cumulative distribution function given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < 3 \\ \frac { 1 } { 6 } ( x - 3 ) ^ { 2 } & 3 \leqslant x < 4 \\ \frac { x } { 3 } - \frac { 7 } { 6 } & 4 \leqslant x < c \\ 1 - \frac { 1 } { 6 } ( d - x ) ^ { 2 } & c \leqslant x < 7 \\ 1 & x \geqslant 7 \end{array} \right.$$ where \(c\) and \(d\) are constants.

1. Show that \(c = 6\)
2. Find \(\mathrm { P } ( X > 3.5 )\)
3. Find \(\mathrm { P } ( X > 4.5 \mid 3.5 < X < 5.5 )\)

1. The continuous random variable \(X\) has the following cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 1 \\ \frac { 4 } { 15 } ( x - 1 ) & 1 < x \leqslant 2 \\ k \left( \frac { a x ^ { 3 } } { 3 } - \frac { x ^ { 4 } } { 4 } \right) + b & 2 < x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$ where \(k , a\) and \(b\) are constants.
Given that the mode of \(X\) is \(\frac { 8 } { 3 }\)

1. show that \(a = 4\)
2. Find \(\mathrm { P } ( X < 2.5 )\) giving your answer to 3 significant figures.

12 The cumulative distribution function of the continuous random variable \(X\) is given by \(F ( x ) = \begin{cases} 0 & x < 20 , \\ a \left( x ^ { 2 } + b x + c \right) & 20 \leqslant x \leqslant 30 , \\ 1 & x > 30 , \end{cases}\) where \(a\), \(b\) and \(c\) are constants.
You are given that \(\mathrm { P } ( X < 25 ) = \frac { 11 } { 24 }\).

1. Find \(\mathrm { P } ( X > 27 )\).
2. Find the 90th percentile of \(X\).