Square root transformation (Y = √X)

Questions asking to find the PDF or CDF of Y where Y = √X or Y² = X, typically requiring the transformation formula with derivative dy/dx = 1/(2√x).

1 questions · Challenging +1.2

5.03g Cdf of transformed variables
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CAIE Further Paper 4 2020 June Q3
8 marks Challenging +1.2
3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 16 } ( 2 - \sqrt { x } ) & 0 \leqslant x < 1 \\ \frac { 3 } { 16 \sqrt { x } } & 1 \leqslant x \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { E } ( X )\).
    The random variable \(Y\) is such that \(Y = \sqrt { X }\).
  2. Find the probability density function of \(Y\).