Continuous CDF with polynomial pieces

4. A continuous random variable \(X\) has cumulative distribution function $$F ( x ) = \left\{ \begin{array} { l r } 0 & x < 0
\frac { 1 } { 4 } x & 0 \leqslant x \leqslant 1
\frac { 1 } { 20 } x ^ { 4 } + \frac { 1 } { 5 } & 1 < x \leqslant d
1 & x > d \end{array} \right.$$

1. Show that \(d = 2\)
2. Find \(\mathrm { P } ( X < 1.5 )\)
3. Write down the value of the lower quartile of \(X\)
4. Find the median of \(X\)
5. Find, to 3 significant figures, the value of \(k\) such that \(\mathrm { P } ( X > 1.9 ) = \mathrm { P } ( X < k )\)

1. The lifetime of a particular battery, \(T\) hours, is modelled using the cumulative distribution function

$$\mathrm { F } ( t ) = \left\{ \begin{array} { l r } 0 & t < 8
\frac { 1 } { 96 } \left( 74 t - \frac { 5 } { 2 } t ^ { 2 } + k \right) & 8 \leqslant t \leqslant 12
1 & t > 12 \end{array} \right.$$

1. Show that \(k = - 432\)
2. Find the probability density function of \(T\), for all values of \(t\).
3. Write down the mode of \(T\).
4. Find the median of \(T\).
5. Find the probability that a randomly selected battery has a lifetime of less than 9 hours. A battery is selected at random. Given that its lifetime is at least 9 hours,
6. find the probability that its lifetime is no more than 11 hours.

7. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 , & x < 0
k x ^ { 2 } + 2 k x , & 0 \leq x \leq 2
8 k , & x > 2 \end{array} \right.$$

1. Show that \(k = \frac { 1 } { 8 }\).
2. Find the median of \(X\).
3. Find the probability density function \(\mathrm { f } ( x )\).
4. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
5. Write down the mode of \(X\).
6. Find \(\mathrm { E } ( X )\).
7. Comment on the skewness of this distribution.

1. The continuous random variable \(Y\) has cumulative distribution function \(\mathrm { F } ( y )\) given by

$$\mathrm { F } ( y ) = \left\{ \begin{array} { c l } 0 & y < 1
k \left( y ^ { 4 } + y ^ { 2 } - 2 \right) & 1 \leqslant y \leqslant 2
1 & y > 2 \end{array} \right.$$

1. Show that \(k = \frac { 1 } { 18 }\).
2. Find \(\mathrm { P } ( Y > 1.5 )\).
3. Specify fully the probability density function f(y).

2. The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0
\frac { 1 } { 4 } \left( y ^ { 3 } - 4 y ^ { 2 } + k y \right) & 0 \leqslant y \leqslant 2
1 & y > 2 \end{array} \right.$$ where \(k\) is a constant.

1. Find the value of \(k\).
2. Find the probability density function of \(Y\), specifying it for all values of \(y\).
3. Find \(\mathrm { P } ( Y > 1 )\).

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1
\frac { x ^ { 3 } } { 10 } + \frac { 3 x ^ { 2 } } { 10 } + a x + b & 1 \leqslant x \leqslant 2
1 & x > 2 \end{array} \right.$$ where \(a\) and \(b\) are constants.

1. Find the value of \(a\) and the value of \(b\).
2. Show that \(\mathrm { f } ( x ) = \frac { 3 } { 10 } \left( x ^ { 2 } + 2 x - 2 \right) , \quad 1 \leqslant x \leqslant 2\)
3. Use integration to find \(\mathrm { E } ( X )\).
4. Show that the lower quartile of \(X\) lies between 1.425 and 1.435

5. The continuous random variable \(T\) represents the time in hours that students spend on homework. The cumulative distribution function of \(T\) is $$\mathrm { F } ( t ) = \begin{cases} 0 , & t < 0
k \left( 2 t ^ { 3 } - t ^ { 4 } \right) & 0 \leq t \leq 1.5
1 , & t > 1.5 \end{cases}$$ where \(k\) is a positive constant.

1. Show that \(k = \frac { 16 } { 27 }\).
2. Find the proportion of students who spend more than 1 hour on homework.
3. Find the probability density function \(\mathrm { f } ( t )\) of \(T\).
4. Show that \(\mathrm { E } ( T ) = 0.9\).
5. Show that \(\mathrm { F } ( \mathrm { E } ( T ) ) = 0.4752\). A student is selected at random. Given that the student spent more than the mean amount of time on homework,
6. find the probability that this student spent more than 1 hour on homework.

8 The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < - 1
\frac { x + 1 } { k + 1 } & - 1 \leqslant x \leqslant k
1 & x > k \end{array} \right.$$ where \(k\) is a positive constant.

1. Find, in terms of \(k\), an expression for \(\mathrm { P } ( X < 0 )\).
2. Determine an expression, in terms of \(k\), for the lower quartile, \(q _ { 1 }\).
3. Show that the probability density function of \(X\) is defined by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { k + 1 } & - 1 \leqslant x \leqslant k
   0 & \text { otherwise } \end{array} \right.$$
4. Given that \(k = 11\) :
   1. sketch the graph of f;
   2. determine \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\);
   3. show that \(\mathrm { P } \left( q _ { 1 } < X < \mathrm { E } ( X ) \right) = 0.25\).

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

$$\begin{array} { l l } \mathrm { F } ( x ) = 0 & x < 2 ,
\mathrm {~F} ( x ) = k ( x - a ) ^ { 2 } & 2 \leq x \leq 6 ,
\mathrm {~F} ( x ) = 1 & x \geq 6 . \end{array}$$

1. Find the values of the constants \(a\) and \(k\).
2. Show that the median of \(X\) is \(2 ( 1 + \sqrt { 2 } )\).
3. Given that \(X > 4\), find the probability that \(X > 5\).

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

$$F ( x ) = \begin{cases} 0 , & x < 2
k \left( 19 x - x ^ { 2 } - 34 \right) , & 2 \leq x \leq 5
1 , & x > 5 \end{cases}$$

1. Show that \(k = \frac { 1 } { 36 }\).
2. Find \(\mathrm { P } ( X > 4 )\).
3. Find and specify fully the probability density function \(\mathrm { f } ( x )\) of \(X\).

4 The cumulative distribution function of the continuous random variable \(X\) is given by
\(\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 ,
k \left( 12 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 2 ,
1 & x > 2 , \end{cases}\) where \(k\) is a constant.

1. Show that \(k = 0.05\).
2. Find \(\mathrm { P } ( 1 \leqslant X \leqslant 1.5 )\).
3. Find the median of \(X\), correct to 3 significant figures.
4. Find which of the median, mean and mode of \(X\) is the largest of the three measures of central tendency.

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 3
c - 4.5 x ^ { n } & 3 \leqslant x \leqslant 9
1 & x > 9 \end{array} \right.$$ where \(c\) is a positive constant and \(n\) is an integer.

1. Showing all stages of your working, find the value of \(c\) and the value of \(n\)
2. Find the lower quartile of \(X\)