Mathematical Methods — Question 17

QCAA 2025, Paper 2 · 4 marks

Q17 · 2025 · Technology-activeComplex familiar4 marks

QUESTION 17 (4 marks)

The probability density function approximates the time (s) that people spend viewing a particular photo on a social media page.

p(t)=\begin{cases}ke^{-7t/50},&0\le t\le20,\\0,&\text{otherwise.}\end{cases}

Determine the mean time people spend viewing the photo.

WORKED SOLUTION

4 marks

ANSWER

5.85

s (approximately 6 s).

Worked solution

Because this is a probability density function,

1=\int_0^{20}ke^{-7t/50}\,dt.

Using a GDC,

k=0.1491.

From the formula for the mean of a continuous random variable,

E(X)=\int_{-\infty}^{\infty}xp(x)\,dx,

so here

E(X)=\int_0^{20}t\,k e^{-7t/50}\,dt=5.85\approx6\text{ s}.

correctly equates the probability density function to 1 over the appropriate domain

[1 mark]

correctly determines the value of

k

[1 mark]

substitutes the domain and developed function

p(t)

into the formula for the mean of a continuous random variable

[1 mark]

determines the mean time

[1 mark]

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

How many marks did you earn?

Compare your working with the guide above.