Methods2025 Exam 1 · Q8Tech-free5 marks

Linear probability density function and transformed integral

Consider

$f(x)=\begin{cases}\frac{3}{8}(4-3x) & 0\le x\le \frac{4}{3}\\ 0 & \text{otherwise}\end{cases}$

The continuous random variable $X$ has probability density function $f(x)$ .

Find $k$ such that $\Pr(X>k)=\frac{9}{16}$ .

3 marks

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The function $h(x)$ is a transformation of $f(x)$ such that

$h(x)=mf(x)+n$

where $m$ and $n$ are real numbers.

Find $\int_0^{\frac{4}{3}} h(x)\,dx$ in terms of $m$ and $n$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.