Transform exponential random variable

9 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 2
a \mathrm { e } ^ { - ( x - 2 ) } & x \geqslant 2 \end{cases}$$ where \(a\) is a constant. Show that \(a = 1\). Find the distribution function of \(X\) and hence find the median value of \(X\). The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\). Find

1. the probability density function of \(Y\),
2. \(\mathrm { P } ( Y > 10 )\).

10 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 0 ,
\frac { a } { 2 ^ { x } } & x \geqslant 0 , \end{cases}$$ where \(a\) is a positive constant. By expressing \(2 ^ { x }\) in the form \(\mathrm { e } ^ { k x }\), where \(k\) is a constant, show that \(X\) has a negative exponential distribution, and find the value of \(a\). State the value of \(\mathrm { E } ( X )\). The variable \(Y\) is related to \(X\) by \(Y = 2 ^ { X }\). Find the distribution function of \(Y\) and hence find its probability density function.