Question 17
The probability density function approximates the time (s) that people spend viewing a particular photo on a social media page.
Determine the mean time people spend viewing the photo.
[4 marks]Stretch the curve until the area is 1, then find where it balances
The shape of the curve is fixed. Only its height $k$ is missing, and the mean can’t be found until the area is exactly 1.
The total area fixes $k$. Then the mean is the balance point, and it sits well left of the middle because the density is highest near $t=0$.
The mean integral only works once the area is 1
Change $k$ and watch two things. The shaded area has to be 1. The triangle shows what $\int_0^{20}t\,p(t)\,dt$ gives with your $k$, and it only lands on the true balance point when the area is right.
The area is 1, so this is a probability density function. Now the mean integral lands exactly on the balance point: $E(T)\approx5.85$ s.
$k=\frac{7}{50}$ would make the area 1 if the times ran on forever. This density stops at 20 s, so the area is only about 0.94 and the mean integral gives about 5.49 instead of 5.85. Integrate over $0\le t\le20$.
The area is less than 1, so this is not a probability density function yet. The mean integral comes out short of the balance point. Increase $k$.
The area is more than 1, so this is not a probability density function. The mean integral overshoots the balance point. Decrease $k$.
The balance point never moves, because $k$ doesn’t change the shape. The mean integral is $39.23k$, so it is only right for the one $k$ that makes the area 1.
Work through the solution one mark at a time
Try each step yourself before you reveal it. Your calculator does both integrals. Half the marks are for setting them up correctly.
The question never asks for $k$, but the mean can’t be found without it. This is a probability density function, so the total area under it is 1. Outside $0\le t\le20$ the function is 0, so integrate over that interval only.
$$1=\int_0^{20}ke^{-\frac{7}{50}t}\,dt$$
Marker: correctly equates the probability density function to 1 over the appropriate domain
$k$ is a constant, so take it outside the integral. Use your calculator for the definite integral, then divide.
$$\begin{aligned}1&=k\int_0^{20}e^{-\frac{7}{50}t}\,dt\\1&\approx6.7085k\\k&\approx0.1491\end{aligned}$$
Exactly, $k=\dfrac{7}{50\left(1-e^{-\frac{14}{5}}\right)}$. Store the full value in your calculator for the next step.
Marker: correctly determines the value of $k$
Take the mean formula from the formula book. Swap in $t$ for the variable, the density with your value of $k$, and the limits 0 and 20.
$$\begin{aligned}E(X)&=\int_{-\infty}^{\infty}x\,p(x)\,dx\\&=\int_0^{20}t\times k\times e^{-\frac{7}{50}t}\,dt\end{aligned}$$
Marker: substitutes domain and developed function $p(t)$ into the formula for the mean of a continuous random variable
Evaluate the integral on your calculator.
$$\begin{aligned}E(X)&=5.85\\&\approx6\text{ s}\end{aligned}$$
Check it makes sense. The density is highest at $t=0$ and falls away, so the mean should sit well below the middle of the interval, which is 10 s.
Marker: determines the mean time
Each phrase of the question gave you something
The question asks for one thing, the mean, but you can’t start there. You have to notice that $k$ is unknown, find it from the total area, and carry it into a second integral. Two standard procedures are joined, and the first one is never asked for. It is still complex familiar because “probability density function” tells you the area is 1, and “mean” sends you to the formula book.
If I had to guess where you might lose marks, it would be the first two. If you go straight to the mean without $k$, the integral gives about 39.23, which is longer than the 20 s the density allows. If you recognise the exponential shape and integrate to infinity, you get $k=\frac{7}{50}$, which misses the first mark’s “appropriate domain”. Keep both integrals on $0\le t\le20$, and check that your mean sits between 0 and 10.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2025 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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