Normalise a density then find its CDF value

Practise normalise a density then find its cdf value with an original VectorPaths question, worked explanation and related VCE Methods practice.

4 practice marks

Original practice question

A density is $f(x)=kx^2$ on $0\leq x\leq2$ and zero elsewhere. (a) Find k. [2 marks] (b) Find $P(X\leq1)$. [2 marks]

Worked explanation

• Part (a)

  $1=\int_0^2kx^2\,dx=8k/3$, so $k=3/8$. The resulting density is nonnegative.

• Part (b)

  $P(X\leq1)=\int_0^1(3/8)x^2\,dx=[x^3/8]_0^1=1/8$.

(a) $3/8$.; (b) $1/8$.

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