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 $0\leq x\leq 2$ and zero elsewhere. Find $k$, $P(X\leq 1)$ and the median.

Worked explanation

• Normalise the density

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

• Integrate to the requested boundary

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

• Solve for the median

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

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

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