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6 The random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 6 } \mathrm { e } ^ { - \frac { 1 } { 6 } x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$ Find

1. the distribution function of \(X\),
2. the probability that \(X\) lies between the median and the mean.

7 The random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$

1. Find the distribution function of \(X\).
2. Find \(\mathrm { P } ( X > 2 )\).
3. Find the median of \(X\).

3 The random variable \(X\) has an exponential distribution with probability density function \(\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x }\) where \(x \geq 0\) 3

1. Show that the cumulative distribution function, for \(x \geq 0\), is given by \(\mathrm { F } ( x ) = 1 - \mathrm { e } ^ { - \lambda x }\)
   [0pt] [3 marks]
   3
2. Given that \(\lambda = 2\), find \(\mathrm { P } ( X > 1 )\), giving your answer to three decimal places.