Continuous Random Variables (Cambridge (CIE) A Level Maths: Probability & Statistics 2): Exam Questions

The continuous random variable $X$ has probability density function $\text{f}$ given by

$\text{f}\left(x\right)=\{\begin{array}{cc}k\left(x^2+3\right)& 3\le x\le 9\\ 0& \text{otherwise}\end{array}$

where $k$ is a constant.

Show that $k=\frac{1}{252}$.

Find $\text{P}\left(X<6\right)$.

The median, $m$, of a continuous random variable $X$, is the value for $X$ which splits the area under the graph of the probability density function exactly in half, such that $\text{P}\left(X<m\right)=0.5$.

Find the value of the median of the continuous random variables which have probability density functions shown in the graphs below.

(i)

(ii)

(iii)

(iv)

State, with a reason, whether each of the following could be graphs of probability density functions.

(i)

(ii)

(iii)

(iv)

This is a semicircle.

State, with a reason, whether the following functions could represent probability density functions.

(i) $\text{f}\left(x\right)=\{\begin{array}{cc}3\left(x^5-2x^3\right)& 1\le x\le 2\\ 0& \text{otherwise}\end{array}$

(ii) $\text{f}\left(x\right)=\{\begin{array}{cc}\frac{3}{8}{\left(x+1\right)}^2& -1\le x\le 1\\ 0& \text{otherwise}\end{array}$

(iii) $\text{f}\left(x\right)=\{\begin{array}{cc}\frac{6}{x^2}& -3\le x\le -2\\ 0& \text{otherwise}\end{array}$

The continuous random variable $X$ has probability density function $\text{f}$ given by

$\text{f}\left(x\right)=\{\begin{array}{cc}k\left(3x^4-2x\right)& 1\le x\le 2\\ 0& \text{otherwise}\end{array}$

where $k$ is a constant.

Show that $k=\frac{5}{78}$.

(i) Find $\text{P}\left(1.2<X<1.8\right)$.

(ii) Find $\text{P}\left(X<1.5\right)$.

The continuous random variable $X$ has probability density function $\text{f}$ given by

$\text{f}\left(x\right)=\{\begin{array}{cc}k\left(2+\sqrt{x}\right)& 4\le x\le 9\\ 0& \text{otherwise}\end{array}$

where $k$ is a constant.

Show that $k=\frac{3}{68}$.

Find $\text{E}\left(X\right)$.

Find $\text{Var}\left(X\right)$.

The continuous random variable $X$ has probability density function $\text{f}$ given by

$\text{f}\left(x\right)=\{\begin{array}{cc}\frac{1}{2}\text{sin}x& 0\le x\le \pi \\ 0& \text{otherwise}\end{array}$

Verify that $\text{f}$ can be a probability density function.

State the value of $\text{E}\left(X\right)$, giving a reason for your answer.

Find the lower quartile of $X$, giving your answer in terms of $\pi$.

The continuous random variable $X$ has probability density function $\text{f}$ given by

$\text{f}\left(x\right)=\{\begin{array}{cc}\frac{5}{62}{x}^{\frac{3}{2}}& 1\le x\le 4\\ 0& \text{otherwise}\end{array}$

Show that the median, $m$, of $X$ satisfies the equation $2m^{\frac{5}{2}}-33=0$.

Hence find the value of $m$.

The continuous random variable $X$ has probability density function $\text{f}$ given by

$\text{f}\left(x\right)=\{\begin{array}{cc}k& 3\le x\le 28\\ 0& \text{otherwise}\end{array}$

where $k$ is a constant.

Find the value of $k$, and find $\text{P}\left(X<15\right)$ and $\text{P}\left(20<X<30\right)$.

(i) State the value of $\text{E}\left(X\right)$.

(ii) Use integration to find $\text{Var}\left(X\right)$.

May calls her nephew Peter each day. Peter's phone ends every call after 15 minutes. The length of a call, $T$ minutes, is modelled by the probability density function given by

$\text{f}\left(t\right)=\{\begin{array}{cc}\frac{1}{1125}{t}^2& 0\le t\le 15\\ 0& \text{otherwise}\end{array}$

Find the mean length of a call.

Find the standard deviation of $T$.

Find the probability that a randomly chosen call lasts longer than 10 minutes.

Out of the next 100 calls, estimate the number that last less than 10 minutes.

The mass, $M$ kilograms, of a wrestler at a local club is modelled by the probability density function given by

$\text{f}\left(m\right)=\{\begin{array}{cc}k-\frac{1}{9000}{\left(m-80\right)}^2& 70\le m\le 100\\ 0& \text{otherwise}\end{array}$

where $k$ is a constant.

Show that $k=\frac{2}{45}$.

Wrestlers who weigh less than 85 kg can enter Heavy Middleweight competitions.

Find the probability that a randomly chosen wrestler can enter Heavy Middleweight competitions.

Wrestlers who weigh less than 80 kg can enter Middleweight competitions.

Given that a wrestler can enter Heavy Middleweight competitions, find the probability that they can also enter Middleweight competitions.

A random variable $X$ takes values between 0 and $k$ only, where $k$ is a constant. The diagram shows the graph of the probability density function, $\text{f}$, of $X$. Between these two values the graph is a straight line.

Find the value of $k$.

Find $\text{P}\left(X>0.4\right)$.

Find the median of $X$.

State, with a reason, whether the following functions could represent probability density functions.

(i) $\text{f}\left(x\right)=\{\begin{array}{cc}\frac{1}{3}\left(x^2-2\right)& 1\le x\le 2\\ 0& \text{otherwise}\end{array}$

(ii) $\text{f}\left(x\right)=\{\begin{array}{cc}\frac{4}{x^5}& x\ge 1\\ 0& \text{otherwise}\end{array}$

(iii) $\text{f}\left(x\right)=\{\begin{array}{cc}\frac{2}{x^2}& -1\le x\le 2\\ 0& \text{otherwise}\end{array}$

The continuous random variable $X$ has probability density function $\text{f}$ given by

$\text{f}\left(x\right)=\{\begin{array}{cc}\frac{1}{4}x\left(x-1\right)\left(x+1\right)& 1\le x\le k\\ 0& \text{otherwise}\end{array}$

where $k$ is a constant.

Find the exact value of $k$.

Find $\text{P}\left(X<\text{E}\left(X\right)\right)$.

Harry draws a straight line by eye, aiming for a length of 10 cm. The length of his line, $L$ cm, is modelled by the probability density function given by

$\text{f}\left(l\right)=\{\begin{array}{cc}\frac{20}{421}{\left(l-10.5\right)}^4& 8\le l\le 12\\ 0& \text{otherwise}\end{array}$

Find the lower quartile of the lengths of Harry's lines.

Harry draws 80 of these lines, independently.

Find the expected number of lines that are shorter than the lower quartile.

The diagram shows the probability density function, $\text{f}\left(x\right)$, of a random variable $X$. For $0\le x\le a$, $\text{f}\left(x\right)=k$; elsewhere $\text{f}\left(x\right)=0$.

Express $k$ in terms of $a$.

State $\text{E}\left(X\right)$ in terms of $a$.

Show that $\text{Var}\left(X\right)=\frac{a^2}{12}$.

Given that $\text{P}\left(X>5\right)=0.6$, find the value of $a$.

The continuous random variable $X$ has probability density function $\text{f}$ given by

$\text{f}\left(x\right)=\{\begin{array}{cc}\frac{1}{2}\left(\text{sin}x+\text{cos}x\right)& 0\le x\le \frac{1}{2}\pi \\ 0& \text{otherwise}\end{array}$

Verify that $\text{f}$ can be a probability density function.

Use integration to show that $\text{E}\left(X\right)=\frac{\pi }{4}$.

Find the exact value of $\text{P}\left(X<\frac{1}{6}\pi \right)$.

The diagram shows the graph of the probability density function, $\text{f}$, of a random variable $T$.

It is given that

$\text{f}\left(t\right)=\{\begin{array}{cc}\frac{3}{32}\left(10t-t^2-21\right)& a\le t\le b\\ 0& \text{otherwise}\end{array}$

Find the values of $a$ and $b$.

State the value of $\text{E}\left(T\right)$ and find $\text{Var}\left(T\right)$.

Find $\text{P}\left(-2\le T\le 5\right)$.

Given that $\text{P}\left(4\le T\le 6\right)=\frac{11}{16}$, find $\text{P}\left(T\ge 6\right)$.

Geoff takes part in a quiz in which he has 4 seconds to answer each question. The time, $T$ seconds, that Geoff takes to answer a question is modelled by the probability density function $\text{f}$ shown in the diagram.

Find the probability that Geoff answers a question in less than 2 seconds.

Find the mean time that Geoff takes to answer a question.

There are 10 questions in the quiz. Assuming that the times are independent, find the probability that Geoff takes longer than 3 seconds to answer at least one of the questions.

The diagram shows the graph of the probability density function, $\text{f}$, of a random variable $X$ which takes values between 0 and $k$ only, where $k$ is a constant. Between these two values the graph is a straight line.

Find the value of $k$.

Find $\text{P}\left(X\le 3\right)$.

Find the interquartile range of $X$.

The time, $T$ seconds, that a computer takes to start up is modelled by the probability density function $\text{f}$ shown in the diagram.

It is given that

$\text{f}\left(t\right)=\{\begin{array}{cc}\frac{1}{8\pi }\sqrt{12t-t^2-20}& a\le t\le b\\ 0& \text{otherwise}\end{array}$

where $a$ and $b$ are constants.

Find the values of $a$ and $b$.

State the mean time for the computer to start up.

Find the probability that the computer takes between 6 and 15 seconds to start up.

Given that $\text{P}\left(4\le T\le 8\right)=\frac{\sqrt{3}}{2\pi }+\frac{1}{3}$, find the exact value of $\text{P}\left(T>8\right)$.

The diagram shows the graph of the probability density function, $\text{g}$, of a random variable $Y$ which takes values between $k$ and 8 only, where $k$ is a constant. Between these two values the graph is a straight line.

Find the value of $k$.

Find $\text{P}\left(5<Y<6\right)$.

Find the median of $Y$.

Find $\text{E}\left(Y\right)$.

The length of a meeting at a company, $T$ hours, is modelled by the probability density function given by

$\text{f}\left(t\right)=\{\begin{array}{cc}\frac{3}{4}t{\left(t-2\right)}^2& 0\le t\le 2\\ 0& \text{otherwise}\end{array}$

Find the mean length of a meeting and find the standard deviation of the lengths.

Show that the median length of a meeting is 0.771 hours, correct to 3 significant figures.

The cost, $C$ euros, of a baguette in France is modelled by the probability density function given by

$\text{f}\left(c\right)=\{\begin{array}{cc}\frac{1}{36}\left(14c-c^2-40\right)& 4\le c\le 10\\ 0& \text{otherwise}\end{array}$

Sketch the graph of $y=\text{f}\left(c\right)$.

Hence explain, without calculation, why the mean cost of a baguette is 7 euros.

Given that $\text{P}\left(C<9\right)=\frac{25}{27}$, find $\text{P}\left(5<C<9\right)$.

The continuous random variable $U$ has probability density function $\text{g}$ given by

$\text{g}\left(u\right)=\{\begin{array}{cc}k& a\le u\le b\\ 0& \text{otherwise}\end{array}$

where $a$, $b$ and $k$ are constants with $b>a$.

(i) Express $k$ in terms of $a$ and $b$.

(ii) State $\text{E}\left(U\right)$ in terms of $a$ and $b$.

(iii) Show by integration that $\text{Var}\left(U\right)=\frac{{\left(b-a\right)}^2}{12}$.

Given that $\text{E}\left(U\right)=8.5$ and $\text{Var}\left(U\right)=6.75$, find the values of $a$ and $b$.

The continuous random variable $X$ has probability density function $\text{f}$ given by

$\text{f}\left(x\right)=\{\begin{array}{cc}\frac{k}{x^3}& 2\le x\le a\\ 0& \text{otherwise}\end{array}$

where $a$ and $k$ are constants.

Show that $k=\frac{8a^2}{a^2-4}$.

Given that $\text{E}\left(X\right)=\frac{20}{7}$, show that $a=5$ and hence find the value of $k$.

Find the exact value of $\text{Var}\left(X\right)$.