Mathematical Methods — Question 13

QCAA 2021, Paper 2 · 7 marks

Q13 · 2021 · Technology-activeSimple familiar7 marks

QUESTION 13 (7 marks)

The amount of gravel (in tonnes) sold by a construction company in a given week is a continuous random variable

X

and has a probability density function defined by

f(x)=\begin{cases}c(1-x^2),&0\le x\le1,\\0,&\text{otherwise.}\end{cases}

Show that

c=\dfrac32

[1 mark]

Determine

P(X<0.25)

[2 marks]

Determine the variance of

X

[4 marks]

WORKED SOLUTION

7 marks

ANSWER

c=\dfrac32

; b)

0.367

; c)

0.059

tonnes

^2

Worked solution

\int_0^1\frac32(1-x^2)\,dx=1.

P(X<0.25)=\int_0^{0.25}\frac32(1-x^2)\,dx=0.367.

E(X)=\int_0^1x\frac32(1-x^2)\,dx=0.375.

Then

\operatorname{Var}(X)=\int_0^1(x-0.375)^2\frac32(1-x^2)\,dx=0.059\text{ tonnes}^2.

correctly establishes the definite integral equated to 1

[1 mark]

correctly establishes the definite integral

[1 mark]

correctly determines the probability

[1 mark]

correctly establishes the definite integral for the mean

[1 mark]

correctly establishes the mean

[1 mark]

establishes definite integral for the variance

[1 mark]

determines variance

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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