Standard +0.3 This is a straightforward application of standard techniques: finding a CDF by integration of a uniform distribution, then transforming it using the relationship Y = e^X. The final part requires solving P(Y ≥ k) = 0.25 using the derived CDF. While it involves multiple steps and the transformation of random variables (a Further Maths topic), each step follows directly from learned procedures without requiring novel insight or complex problem-solving.
9 The continuous random variable \(X\) has probability density function f given by
$$f ( x ) = \begin{cases} \frac { 1 } { 2 a } & - a \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$
where \(a\) is a positive constant. Find the distribution function of \(X\).
The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\). Find the distribution function of \(Y\).
Given that \(a = 4\), find the value of \(k\) for which \(\mathrm { P } ( Y \geqslant k ) = 0.25\).
9 The continuous random variable $X$ has probability density function f given by
$$f ( x ) = \begin{cases} \frac { 1 } { 2 a } & - a \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$
where $a$ is a positive constant. Find the distribution function of $X$.
The random variable $Y$ is defined by $Y = \mathrm { e } ^ { X }$. Find the distribution function of $Y$.
Given that $a = 4$, find the value of $k$ for which $\mathrm { P } ( Y \geqslant k ) = 0.25$.
\hfill \mbox{\textit{CAIE FP2 2012 Q9 [9]}}