Cumulative distribution functions

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

1. Find the distribution function of \(X\).
2. Find the exact value of the interquartile range of \(X\).

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 \(X\).

2 The continuous random variable \(U\) has (cumulative) distribution function given by $$\mathrm { F } ( u ) = \begin{cases} \frac { 1 } { 5 } \mathrm { e } ^ { u } & u < 0
1 - \frac { 4 } { 5 } \mathrm { e } ^ { - \frac { 1 } { 4 } u } & u \geqslant 0 \end{cases}$$

1. Find the upper quartile of \(U\).
2. Find the probability density function of \(U\).

6 The continuous random variable \(X\) has a uniform distribution on the interval \([ - \pi , \pi ]\).
The random variable \(Y\) is defined by \(Y = \sin X\).
Determine the cumulative distribution function of \(Y\).

6. The discrete random variable \(X\) can take only the values 2,3 or 4 . For these values the cumulative distribution function is defined by $$F ( x ) = \frac { ( x + k ) ^ { 2 } } { 25 } \text { for } x = 2,3,4$$ where \(k\) is a positive integer.

1. Find \(k\).
2. Find the probability distribution of \(X\).

2
\includegraphics[max width=\textwidth, alt={}, center]{43b2498f-73e2-4d33-adaf-fc3e460fa36a-2_358_1093_495_520} A random variable \(X\) takes values between 0 and 4 only and has probability density function as shown in the diagram. Calculate the median of \(X\).

7 The continuous random variable \(X\) has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0
1 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } & x \geqslant 0 \end{cases}$$ For a random value of \(X\), find the probability that 2 lies between \(X\) and \(4 X\). Find also the expected value of the width of the interval ( \(X , 4 X\) ).

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 \(X\).

4. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 2
k \left( a x + b x ^ { 2 } - x ^ { 3 } \right) & 2 \leqslant x \leqslant 3
1 & x > 3 \end{array} \right.$$ Given that the mode of \(X\) is \(\frac { 8 } { 3 }\)

1. show that \(b = 8\)
2. find the value of \(k\).

6 The continuous random variable \(X\) has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.4 x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { P } ( X > 2 )\).
2. Find the interquartile range of \(X\).

1. The three independent random variables \(A , B\) and \(C\) each have a continuous uniform distribution over the interval \([ 0,5 ]\).
   1. Find the probability that \(A , B\) and \(C\) are all greater than 3
   The random variable \(Y\) represents the maximum value of \(A , B\) and \(C\).
   The cumulative distribution function of \(Y\) is $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0
   \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5
   1 & y > 5 \end{cases}$$
2. Using algebraic integration, show that \(\operatorname { Var } ( Y ) = 0.9375\)
3. Find the mode of \(Y\), giving a reason for your answer.
4. Describe the skewness of the distribution of \(Y\). Give a reason for your answer.
5. Find the value of \(k\) such that \(\mathrm { P } ( k < Y < 2 k ) = 0.189\)

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

$$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0
\frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k
\frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6
1 & y > 6 \end{array} \right.$$

1. Find \(\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)\)
2. Find the value of \(k\)
3. Use algebraic calculus to find \(\mathrm { E } ( Y )\)

6 The function \(\mathrm { F } ( t )\) is defined as follows. $$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0
\sin ^ { 4 } t & 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi
1 & t > \frac { 1 } { 2 } \pi \end{cases}$$

1. Verify that F is a (cumulative) distribution function. The continuous random variable \(T\) has (cumulative) distribution function \(\mathrm { F } ( t )\).
2. Find the lower quartile of \(T\).
3. Find the (cumulative) distribution function of \(Y\), where \(Y = \sin T\), and obtain the probability density function of \(Y\).
4. Find the expected value of \(\frac { 1 } { Y ^ { 3 } + 2 Y ^ { 4 } }\).

2 A continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \frac { 1 } { x ^ { 3 } } & x \geqslant 1
0 & \text { otherwise } \end{cases}$$ Find \(E ( \sqrt { } X )\).

7. The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0
2 x ^ { 2 } - x ^ { 3 } , & 0 \leqslant x \leqslant 1
1 , & x > 1 \end{cases}$$

1. Find \(\mathrm { P } ( X > 0.3 )\).
2. Verify that the median value of \(X\) lies between \(x = 0.59\) and \(x = 0.60\).
3. Find the probability density function \(\mathrm { f } ( x )\).
4. Evaluate \(\mathrm { E } ( X )\).
5. Find the mode of \(X\).
6. Comment on the skewness of \(X\). Justify your answer.

7 The continuous random variable \(X\) has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 ,
\frac { 1 } { 90 } \left( x ^ { 2 } + x ^ { 4 } \right) & 0 \leqslant x \leqslant 3 ,
1 & x > 3 . \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 2 }\).

1. Find the probability density function of \(Y\).
2. Find the mean value of \(Y\).

2 The circumferences, \(c \mathrm {~cm}\), of some trees in a wood were measured. The results are summarised in the table.

1. On the grid, draw a cumulative frequency graph to represent the information.
   \includegraphics[max width=\textwidth, alt={}, center]{9c23b94b-e573-4e13-be90-e63a0daf18e5-03_1401_1404_854_413}
2. Estimate the percentage of trees which have a circumference larger than 75 cm .

8 The radius, \(R\), of a sphere is a random variable with a continuous uniform distribution between 0 and 10 .

1. Find the cumulative distribution function and probability density function of \(A\), the surface area of the sphere.
2. Find \(\mathrm { P } ( \mathrm { A } \leqslant 200 \pi )\). \section*{END OF QUESTION PAPER}

1. The cumulative distribution of a discrete random variable \(X\) is given by

where \(k\) is a positive constant.

1. Show that \(k = 4\)
2. Find the probability distribution of the discrete random variable \(X\)
3. Using your answer to part (b), write down the mode of \(X\)
4. Calculate \(\operatorname { Var } ( 13 X - 6 )\)