Exponential transformation (Y = e^X)

Questions asking to find the PDF or CDF of Y where Y involves an exponential function of X (e.g., Y = e^(kX)), requiring logarithmic inverse transformation.

1 questions

CAIE FP2 2017 June Q9
9 The continuous random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 0 ,
a \mathrm { e } ^ { - x \ln 2 } & x \geqslant 0 , \end{cases}$$ where \(a\) is a positive constant.
  1. Find the value of \(a\).
  2. State the value of \(\mathrm { E } ( X )\).
  3. Find the interquartile range of \(X\).
    The variable \(Y\) is related to \(X\) by \(Y = 2 ^ { X }\).
  4. Find the probability density function of \(Y\).