Mathematical Methods — Question 16

QCAA 2022, Paper 2 · 4 marks

Q16 · 2022 · Technology-activeComplex familiar4 marks

QUESTION 16 (4 marks)

The time spent waiting in a queue at a certain supermarket is given by

(X+11)

minutes, where

X

is a random variable with the probability density function

f(x)=\begin{cases}\dfrac{a(4-x^2)}{32},&-2\le x\le2,\\0,&\text{otherwise.}\end{cases}

Determine the probability of waiting between 10 and 12 minutes in a queue at this supermarket.

WORKED SOLUTION

4 marks

ANSWER

0.6875

Worked solution

Normalisation gives

\int_{-2}^{2}\frac{a(4-x^2)}{32}\,dx=1,

a=3

. Waiting between 10 and 12 minutes means

-1\le X\le1

. Hence

P(-1\le X\le1)=\int_{-1}^{1}\frac{3(4-x^2)}{32}\,dx=0.6875.

correctly identifies required integral equation to solve

[1 mark]

correctly determines the value of

a

[1 mark]

correctly identifies interval

[1 mark]

determines probability

[1 mark]

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

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