Continuous random variables: past paper questions

Real questions on continuous random variables

The newest 20 of 61 written questions. Open one to see it on the real paper.

1. 9709/63 · May/June 2026 · Q3[6]

   6-mark question — its maths and diagrams are best seen on the paper.

   See it on the paper →
2. 9709/61 · October/November 2025 · Q7a[4]
   The time, in minutes, taken by students to complete a test is modelled by the random variable $X$ with probability density function $$f(x)=\{\begin{array}{ll}-\frac{3}{4}(x-3)(x-5)& 3\le x\le 5,\\ 0& \text{otherwise}.\end{array}$$ (a) Find the probability that a randomly chosen student takes longer than 4.5 minutes...See it on the paper →
3. 9709/61 · October/November 2025 · Q7b[1]
   The time, in minutes, taken by students to complete a test is modelled by the random variable $X$ with probability density function $$f(x)=\{\begin{array}{ll}-\frac{3}{4}(x-3)(x-5)& 3\le x\le 5,\\ 0& \text{otherwise}.\end{array}$$ (b) Write down the median of $X$.See it on the paper →
4. 9709/61 · October/November 2025 · Q7c[2]
   The time, in minutes, taken by students to complete a test is modelled by the random variable $X$ with probability density function $$f(x)=\{\begin{array}{ll}-\frac{3}{4}(x-3)(x-5)& 3\le x\le 5,\\ 0& \text{otherwise}.\end{array}$$ (c) Without performing an integration, use your answer to part (a) to find $\text{P}(3...See it on the paper →
5. 9709/62 · October/November 2025 · Q5a[3]
   A random variable $X$ has probability density function given by $$f(x)=\{\begin{array}{ll}k(2x^2-x^3)& 0\le x\le 2,\\ 0& \text{otherwise}.\end{array}$$ (a) Show that $k=\frac{3}{4}$.See it on the paper →
6. 9709/62 · October/November 2025 · Q5b(i)[1]
   A random variable $X$ has probability density function given by $$f(x)=\{\begin{array}{ll}k(2x^2-x^3)& 0\le x\le 2,\\ 0& \text{otherwise}.\end{array}$$ (b) (i) Write down the value of $\text{P}(X\le m)$.See it on the paper →
7. 9709/62 · October/November 2025 · Q5b(ii)[5]
   A random variable $X$ has probability density function given by $$f(x)=\{\begin{array}{ll}k(2x^2-x^3)& 0\le x\le 2,\\ 0& \text{otherwise}.\end{array}$$ (ii) Hence find $\text{P}(\text{E}(X)\le X\le m)$.See it on the paper →
8. 9709/62 · May/June 2025 · Q7a[3]
   The random variable $X$ has probability density function given by $$f(x)=\{\begin{array}{ll}\frac{k{x}^2}{a^2}& 0\le x\le a,\\ 0& \text{otherwise,}\end{array}$$ where $k$ and $a$ are positive constants. (a) Show that $k=\frac{3}{a}$.See it on the paper →
9. 9709/62 · May/June 2025 · Q7b[3]
   The random variable $X$ has probability density function given by $$f(x)=\{\begin{array}{ll}\frac{k{x}^2}{a^2}& 0\le x\le a,\\ 0& \text{otherwise},\end{array}$$ where $k$ and $a$ are positive constants. (b) Find the value of $a.$See it on the paper →
10. 9709/62 · May/June 2025 · Q7c[3]
    The random variable $X$ has probability density function given by $$f(x)=\{\begin{array}{ll}\frac{k{x}^2}{a^2}& 0\le x\le a,\\ 0& \text{otherwise},\end{array}$$ where $k$ and $a$ are positive constants. (c) Find the median of $X$.See it on the paper →
11. 9709/63 · May/June 2025 · Q6b[6]
    A random variable $X$ has probability density function given by $$f(x)=\{\begin{array}{ll}ax& 0\le x\le b,\\ 0& \text{otherwise,}\end{array}$$ where $a$ and $b$ are constants. (b) Show that $P(X<E(X))=\frac{4}{9}$.See it on the paper →
12. 9709/61 · October/November 2024 · Q4b[3]
    A random variable $X$ has probability density function f defined by $$f(x)=\{\begin{array}{ll}\frac{a}{x^2}-\frac{18}{x^3}& 2\le x\le 3,\\ 0& \text{otherwise},\end{array}$$ where $a$ is a constant. (b) Show that $E(X)=\frac{27}{2}\ln \frac{3}{2}-3$.See it on the paper →
13. 9709/62 · October/November 2024 · Q5c[6]
    A machine puts sweets into bags at random. The numbers of lemon and orange sweets in a bag have the independent distributions Po(3.7) and Po(2.6) respectively. A bag of sweets is chosen at random. (c) Use approximating distributions to find the probability that the total number of lemon sweets in the 10 bags is less th...See it on the paper →
14. 9709/62 · October/November 2024 · Q6a[1]
    The time, $X$ hours, taken by a large number of people to complete a challenge is modelled by the probability density function given by $$f(x)=\{\begin{array}{ll}\frac{1}{x^2}& a⩽x⩽b,\\ 0& \text{otherwise},\end{array}$$ where $a$ and $b$ are constants. (a) State what the constants $a$ and $b$ repre...See it on the paper →
15. 9709/62 · October/November 2024 · Q6b[3]
    The time, $X$ hours, taken by a large number of people to complete a challenge is modelled by the probability density function given by $$f(x)=\{\begin{array}{ll}\frac{1}{x^2}& a⩽x⩽b,\\ 0& \text{otherwise},\end{array}$$ where $a$ and $b$ are constants. (b) Show that $a=\frac{b}{b+1}$.See it on the paper →
16. 9709/62 · October/November 2024 · Q6c[4]
    The time, $X$ hours, taken by a large number of people to complete a challenge is modelled by the probability density function given by $$f(x)=\{\begin{array}{ll}\frac{1}{x^2}& a⩽x⩽b,\\ 0& \text{otherwise},\end{array}$$ where $a$ and $b$ are constants. (c) Show that $b=2$ and find the value of $a...See it on the paper →
17. 9709/62 · October/November 2024 · Q6d[3]
    The time, $X$ hours, taken by a large number of people to complete a challenge is modelled by the probability density function given by $$f(x)=\{\begin{array}{ll}\frac{1}{x^2}& a\le x\le b,\\ 0& \text{otherwise},\end{array}$$ where $a$ and $b$ are constants. (d) Find the median of $X$.See it on the paper →
18. 9709/61 · May/June 2024 · Q6c[4]
    [Diagram: A graph in the first quadrant with axes $x$ and $f(x)$ intersecting at $O$. A quarter circle arc connects the point $(0,a)$ on the $f(x)$-axis to the point $(a,0)$ on the $x$-axis.] The diagram shows the graph of the probability density function, $f$, of a random variable $X$. The graph is a quarter circle...See it on the paper →
19. 9709/62 · February/March 2024 · Q4a[5]
    Each year a transport firm uses $X$ litres of gasoline and $Y$ litres of diesel fuel, where $X$ and $Y$ have the independent distributions $X\sim \mathrm{N}(10700,950^2)$ and $Y\sim \mathrm{N}(13400,1210^2)$. (a) Find the probability that in a randomly chosen year the firm uses more gasoline than diesel fuel. T...See it on the paper →
20. 9709/62 · February/March 2024 · Q4b[5]
    Each year a transport firm uses $X$ litres of gasoline and $Y$ litres of diesel fuel, where $X$ and $Y$ have the independent distributions $X\sim \mathrm{N}(10700,950^2)$ and $Y\sim \mathrm{N}(13400,1210^2)$. (b) Find the probability that the total cost of gasoline and diesel fuel in a randomly chosen year is b...See it on the paper →

What examiners said

A sketch of a possible curve for the pdf was helpful here to show the positions of p and 1.3p, as well as the symmetrical positions of equal values on the right-hand side of the pdf. Many different approaches to find the relevant areas were possible. One direct way was to sum the three adjacent areas of 1.3p, 1.3p and p from the sketch.

— Examiner report, October/November 2023, 9709/63 Q6(b) (Cambridge)

In this part, while the use of limits from a to 2 was expected, other limits and values could be used. The integration and substitution led to the cubic equation a3 – 9a2+ 8 = 0 or an equivalent form. A complete solution could be found most simply by factorising to (a – 1)(a2 – 8a – 8). The two solutions from the quadratic were outside the range of the pdf and so not relevant, leaving the sole ans...

— Examiner report, October/November 2023, 9709/63 Q6(c) (Cambridge)

The majority of candidates scored part marks on this question with very few achieving full marks. Many candidates used E(X – Y) = 1 and Var(X – Y) = 5 or equivalent (though some subtracted the variances using 3 – 2 rather than 3 + 2). Many also successfully calculated the probability of (X – Y) > 2 but did not identify the other possible case of (Y – X) > 2 i.e. (X – Y) < –2. Of those who did find...

— Examiner report, May/June 2023, 9709/62 Q5(a) (Cambridge)