Variance of transformed variable

7 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 4 } \left( 4 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Find \(\operatorname { Var } ( \sqrt { X } )\).
   The continuous random variable \(Y\) is defined by \(Y = X ^ { 2 }\).
2. Find the probability density function of \(Y\).
3. Find the exact value of the median of \(Y\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

6 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 x } { 44 } + \frac { 1 } { 22 } & 1 \leq x \leq 5 \\ 0 & \text { otherwise } \end{cases}$$ 6

1. Find \(\mathrm { P } ( X > 2 )\) [0pt] [2 marks]
   6
2. Find the upper quartile of \(X\) Give your answer to two decimal places.
   6
3. Find \(\operatorname { Var } \left( 44 X ^ { - 3 } \right)\) Give your answer to three decimal places.

A continuous random variable \(X\) has probability density function f(\(x\)) where $$f(x) = \begin{cases} kx^n & 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) and \(n\) are positive integers.

1. Find \(k\) in terms of \(n\). [3]
2. Find E(\(X\)) in terms of \(n\). [3]
3. Find E(\(X^2\)) in terms of \(n\). [2]

Given that \(n = 2\)

1. find Var(3\(X\)). [3]