Find parameter from probability condition

5 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x - 1 } & 3 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { \ln 2 }\).
2. Find \(a\) such that \(\mathrm { P } ( X < a ) = 0.75\).

5 The volume, in \(\mathrm { cm } ^ { 3 }\), of liquid left in a glass by people when they have finished drinking all they want is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} k ( x - 2 ) ^ { 2 } & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 8 }\).
2. 20\% of people leave at least \(d \mathrm {~cm} ^ { 3 }\) of liquid in a glass. Find \(d\).
3. Find \(\mathrm { E } ( X )\).

5 The volume, in \(\mathrm { cm } ^ { 3 }\), of liquid left in a glass by people when they have finished drinking all they want is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} k ( x - 2 ) ^ { 2 } & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 8 }\).
2. \(20 \%\) of people leave at least \(d \mathrm {~cm} ^ { 3 }\) of liquid in a glass. Find \(d\).
3. Find \(\mathrm { E } ( X )\).

3 A continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c l } \frac { 3 } { 2 a ^ { 3 } } x ^ { 2 } & - a \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$ where \(a\) is a constant.

1. It is given that \(\mathrm { P } ( - 3 \leqslant X \leqslant 3 ) = 0.125\). Find the value of \(a\) in this case.
2. It is given instead that \(\operatorname { Var } ( X ) = 1.35\). Find the value of \(a\) in this case.
3. Explain the relationship between \(x\) and \(X\) in this question.