Mathematical Methods — Question 14

QCAA 2020, Paper 2 · 6 marks

Q14 · 2020 · Technology-activeSimple familiar6 marks

QUESTION 14 (6 marks)

Let

X

denote the time in minutes between the arrival of trains at a station. The cumulative distribution function of

X

is defined by

F(x)=\begin{cases}2-\frac{10}{x},&5\le x\le10,\\0,&\text{otherwise.}\end{cases}

Determine the probability density function of

X

[3 marks]

Determine the probability that

5<X<7

[1 mark]

Determine the mean time between the arrival of trains at the station.

[2 marks]

WORKED SOLUTION

6 marks

ANSWER

Accepted marking-guide Method 1: a)

f(x)=10/x^2

5\le x\le10

; b)

4/7

; c) 6.93 minutes

Worked solution

Method 1 — interpreting the given equation as a CDF: a) differentiate to obtain

f(x)=\begin{cases}\frac{10}{x^2},&5\le x\le10,\\0,&\text{otherwise.}\end{cases}

F(7)-F(5)=4/7

. c)

E(X)=\int_5^{10}x\frac{10}{x^2}dx=\int_5^{10}\frac{10}{x}dx=6.93\text{ minutes}.

The official marking guide also supplied an alternative accepted method interpreting the printed expression as a PDF:

F(x)=\begin{cases}0,&x<5,\\2\left(x-5\ln(x/5)-5\right),&5\le x\le10,\\1,&x>10,\end{cases}

giving

P(5<X<7)=0.6353

and mean 25 minutes.

correctly identifies the use of the derivative of the cumulative distribution function

[1 mark]

correctly determines the derivative of the cumulative distribution function (pdf)

[1 mark]

uses appropriate convention to communicate the probability density function as a piecewise function with given domains

[1 mark]

correctly determines the probability that there are 5 to 7 minutes between train arrivals

[1 mark]

provides a statement identifying the use of expected value for a continuous random variable

[1 mark]

determines mean time between arrivals

[1 mark]

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

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Compare your working with the guide above.