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.

4. The length of a telephone call made to a company is denoted by the continuous random variable \(T\). It is modelled by the probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c l } k t & 0 \leqslant t \leqslant 10
0 & \text { otherwise } \end{array} \right.$$

1. Show that the value of \(k\) is \(\frac { 1 } { 50 }\).
2. Find \(\mathrm { P } ( T > 6 )\).
3. Calculate an exact value for \(\mathrm { E } ( T )\) and for \(\operatorname { Var } ( T )\).
4. Write down the mode of the distribution of \(T\). It is suggested that the probability density function, \(\mathrm { f } ( t )\), is not a good model for \(T\).
5. Sketch the graph of a more suitable probability density function for \(T\).

6. A random variable \(X\) has a probability density function given by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 4 x ^ { 2 } ( 3 - x ) } { 27 } & 0 \leq x \leq 3
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$

1. Find the mode of \(X\).
2. Find the mean of \(X\).
3. Specify completely the cumulative distribution function of \(X\).
4. Deduce that the median, \(m\), of \(X\) satisfies the equation \(m ^ { 4 } - 4 m ^ { 3 } + 13 \cdot 5 = 0\), and hence show that \(1.84 < m < 1.85\).
5. What do these results suggest about the skewness of the distribution?

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.