CDF with additional constraints

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

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

1. Show that \(a k = 1 - b k ^ { 2 }\) Using part (a) and given that \(\mathrm { E } ( X ) = \frac { 6 } { 5 }\)
2. show that \(5 b k ^ { 3 } = 36 - 15 k\) Using part (a) and given that \(\mathrm { E } ( X ) = \frac { 6 } { 5 }\) and \(\operatorname { Var } ( X ) = \frac { 22 } { 75 }\)
3. show that \(5 b k ^ { 4 } = 52 - 10 k ^ { 2 }\) Given that \(k < 3\)
4. find the value of \(k\)
5. Hence find the value of \(a\) and the value of \(b\)

1. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1
k \left( a x + b x ^ { 3 } - x ^ { 4 } - 4 \right) & 1 \leqslant x \leqslant 2
1 & x > 2 \end{array} \right.$$ where \(a\), \(b\) and \(k\) are non-zero constants.
Given that the mode of \(X\) is 1.5

1. show that \(b = 3\)
2. Hence show that \(a = 2\)
3. Show that the median of \(X\) lies between 1.4 and 1.5

5. The random variable \(X\) has cumulative distribution function given by $$F ( x ) = \left\{ \begin{array} { l r } 0 & x < 0
\frac { 1 } { 100 } \left( a x ^ { 3 } + b x ^ { 2 } + 15 x \right) & 0 \leqslant x \leqslant 5
1 & x > 5 \end{array} \right.$$ Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 6.25\)

1. show that \(6 a + b = 0\)
2. find the value of \(a\) and the value of \(b\)
3. find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\)

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 \(k\).

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 4
p x - k \sqrt { x } & 4 \leqslant x \leqslant 9
1 & x > 9 \end{array} \right.$$ where \(p\) and \(k\) are constants.

1. Find the value of \(p\) and the value of \(k\). Given that \(\mathrm { E } ( X ) = \frac { 119 } { 18 }\)
2. show that \(\operatorname { Var } ( X ) = 2.05\) to 3 significant figures.
3. Write down the mode of \(X\).
4. Find the exact value of the constant \(a\) such that \(\mathrm { P } ( X \leqslant a ) = \frac { 7 } { 27 }\)

4 The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 0
k \left( x ^ { 3 } - \frac { 3 } { 8 } x ^ { 4 } \right) & 0 < x \leqslant 2
1 & x > 2 \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 2 }\)
2. Showing your working clearly, use calculus to find
   1. \(\mathrm { E } ( X )\)
   2. the mode of \(X\)
3. Describe, giving a reason, the skewness of the distribution of \(X\)

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

$$F ( x ) = \left\{ \begin{array} { c r } 0 & x < 0
k \left( x - a x ^ { 2 } \right) & 0 \leqslant x \leqslant 4
1 & x > 4 \end{array} \right.$$ The values of \(a\) and \(k\) are positive constants such that \(\mathrm { P } ( X < 2 ) = \frac { 2 } { 3 }\)

1. Find the exact value of the median of \(X\)
2. Find the probability density function of \(X\)
3. Hence, deduce the value of the mode of \(X\), giving a reason for your answer.

1. A continuous random variable \(X\) has probability density function

$$f ( x ) = \left\{ \begin{array} { c l } a x ^ { - 2 } - b x ^ { - 3 } & 2 \leqslant x < \infty
0 & \text { otherwise } \end{array} \right.$$ where \(a\) and \(b\) are constants. Given that \(\mathrm { P } ( X \leqslant 4 ) = \frac { 3 } { 8 }\)

1. use algebraic integration to show that \(a = 3\) Show your working clearly.
2. Find the exact value of the median of \(X\)