CAIE Further Paper 4 2022 November — Question 5

Exam BoardCAIE
ModuleFurther Paper 4 (Further Paper 4)
Year2022
SessionNovember
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TopicContinuous Probability Distributions and Random Variables
5 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 0 \\ 1 - \frac { 1 } { 144 } ( 12 - x ) ^ { 2 } & 0 \leqslant x \leqslant 12 \\ 1 & x > 12 \end{cases}$$
  1. Find the upper quartile of \(X\).
  2. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\).
    The random variable \(Y\) is given by \(Y = \sqrt { X }\).
  3. Find the probability density function of \(Y\).
5 The continuous random variable $X$ has cumulative distribution function F given by

$$F ( x ) = \begin{cases} 0 & x < 0 \\ 1 - \frac { 1 } { 144 } ( 12 - x ) ^ { 2 } & 0 \leqslant x \leqslant 12 \\ 1 & x > 12 \end{cases}$$

(a) Find the upper quartile of $X$.\\

(b) Find $\operatorname { Var } \left( X ^ { 2 } \right)$.\\

The random variable $Y$ is given by $Y = \sqrt { X }$.\\
(c) Find the probability density function of $Y$.\\

\hfill \mbox{\textit{CAIE Further Paper 4 2022 Q5}}
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Q1 Q2 Q3 Q4 Q5 Q6