Continuous expectation and variance

Practise continuous expectation and variance with an original VectorPaths question, worked explanation and related VCE Methods practice.

4 practice marks

Original practice question

X has density $f(x)=3x^2$ for $0\leq x\leq1$, and zero elsewhere. (a) Find E(X). [2 marks] (b) Find Var(X) exactly. [2 marks]

Worked explanation

• Part (a)

  $E(X)=\int_0^1x(3x^2)\,dx=[3x^4/4]_0^1=3/4$.

• Part (b)

  $E(X^2)=\int_0^1x^2(3x^2)\,dx=3/5$. Thus variance is $3/5-(3/4)^2=3/80$.

(a) $3/4$.; (b) $3/80$.

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