Find mode of distribution

1. The continuous random variable \(Y\) has the following probability density function

$$f ( y ) = \begin{cases} \frac { 6 } { 25 } ( y - 1 ) & 1 \leqslant y < 2 \\ \frac { 3 } { 50 } \left( 4 y ^ { 2 } - y ^ { 3 } \right) & 2 \leqslant y < 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch f(y)
2. Find the mode of \(Y\)
3. Use algebraic integration to calculate \(\mathrm { E } \left( Y ^ { 2 } \right)\) Given that \(\mathrm { E } ( Y ) = 2.696\)
4. find \(\operatorname { Var } ( Y )\)
5. Find the value of \(y\) for which \(\mathrm { P } ( Y \geqslant y ) = 0.9\) Give your answer to 3 significant figures.

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

$$f ( x ) = \left\{ \begin{array} { c c } 0.8 - 6.4 x ^ { - 3 } & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$ The median of \(X\) is \(m\)

1. Show that \(m ^ { 3 } - 3.625 m ^ { 2 } + 4 = 0\)
2. 1. Find \(\mathrm { f } ^ { \prime } ( x )\)
   2. Explain why the mode of \(X\) is 4 Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 10.5\) to 3 significant figures,
3. find \(\operatorname { Var } ( X )\), showing your working clearly.

1 The lifetime, \(T\) years, of a mortgage may be modelled by the random variable \(T\) with probability density function \(\mathrm { f } ( t )\), where $$\mathrm { f } ( t ) = \begin{cases} k \sin \left( \frac { 3 } { 32 } t \right) & 0 \leqslant t \leqslant 8 \pi \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 3 } { 32 } ( 2 - \sqrt { 2 } )\).
2. Sketch the graph of \(\mathrm { f } ( t )\) and state the mode.