normalising a continuous density

Practise normalising a continuous density with an original VectorPaths question, worked explanation and related VCE Methods practice.

3 practice marks

Original practice question

A density is $f(x)=kx$ for $0\leq x\leq 3$ and zero elsewhere. Find $k$, $P(X\leq \frac{3}{2})$ and the median.

Worked explanation

• Normalise the density

  $\int_0^3kx\,dx=ka^2/2=1$ with $a=3$, hence $k=\frac{2}{9}$.

• Integrate to the requested boundary

  $P(X\leq \frac{3}{2})=\int_0^{\frac{3}{2}}\frac{2 x}{9}\,dx=1/4$.

• Solve for the median

  $\int_0^m \frac{2 x}{9}\,dx=1/2$, so $m=\frac{3 \sqrt{2}}{2}$ (the positive solution in the support).

$k=\frac{2}{9}$, probability $=1/4$, median $=\frac{3 \sqrt{2}}{2}$.

Original VectorPaths practice; not an official VCAA question or marking rubric.