Continuous Random Variables (DP IB Analysis & Approaches (AA): HL): Exam Questions

A population of grasshoppers is being studied. The length, in cm, of an adult grasshopper can be modelled by a continuous random variable $X$ with probability density function $f$ given by

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

Find the value of $k$.

Sketch the graph of $y=f\left(x\right)$.

Find the probability that a grasshopper picked at random is less than 4 cm in length.

The continuous random variable $X$ has probability density function $f$ given by

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

Show that $\mathrm{E}\left(X\right)=1$.

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

Hence, find the standard deviation of $X$, giving your answer in the form $\frac{\sqrt{a}}{b}$, where $a,b\in {\mathrm{\mathbb{Z}}}^{+}$.

The continuous random variable $X$ has probability density function $f$ given by

$f\left(x\right)=\{\begin{array}{cc}t{x}^3-\frac{x^2}{18}+\frac{7}{36}x,& 0\le x\le 6\\ 0,& \text{otherwise}\end{array}$

Find the value of $t$.

Find

(i) $\mathrm{E}\left(X\right)$

(ii) the mode of $X$

(iii) the median of $X$.

The continuous random variable $X$ has probability density function $f$ given by

$f\left(x\right)=\{\begin{array}{cc}\frac{1}{10}{x}^2,& 0\le x<2\\ \frac{82}{135}-\frac{14}{135}x,& 2\le x\le 5\\ 0,& \text{otherwise}\end{array}$

Sketch the graph of $y=f\left(x\right)$.

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

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

The continuous random variable $X$ has probability density function $f$ given by

$f\left(x\right)=\{\begin{array}{cc}k(x+4),& 0\le x\le 1\\ k(6-x),& 1<x\le 6\\ 0,& \text{otherwise}\end{array}$

Find the value of $k$.

Sketch the graph of $y=f\left(x\right)$.

Find

(i) $\mathrm{E}\left(X\right)$

(ii) $\mathrm{Var}\left(X\right)$

(iii) $\mathrm{P}(0.5\le X\le 1.5)$.

A UK energy company charges £0.22 per kilowatt hour (kWh) of electricity used.

The amount of electricity, $X$ kWh, used in a day by a randomly chosen customer can be modelled by a continuous random variable with probability density function $f$ given by

$f\left(x\right)=\{\begin{array}{cc}\frac{x(k-x)}{972},& 0\le x\le 18\\ 0,& \text{otherwise}\end{array}$

Show that $k=18$.

A customer’s total daily charge consists of a fixed (standing) charge of £0.38 per day plus the charge for the electricity used.

(i) Find the expected total daily charge.

(ii) Find the standard deviation for the total daily charge.

Consider the function $f$ defined by

$f\left(x\right)=\{\begin{array}{cc}k(x-4){\left(x-5\right)}^2,& 4\le x\le 5\\ 0,& \text{otherwise}\end{array}$

Show that $f$ is a probability density function when $k=12$.

When $k=12$, $f$ is the probability density function of the continuous random variable $X$.

Find the mode of $X$.

Find the median of $X$.

Consider the function $f$ defined by

$f\left(x\right)=\frac{1}{2\pi }\sqrt{4-x^2}$

Sketch the graph of $y=f\left(x\right)$.

Write down the domain of $f$ for which $f$ is the probability density function of a continuous random variable $X$.

(i) Find $\mathrm{P}(X<-1)$.

(ii) Write down $\mathrm{E}\left(X\right)$.

(iii) Write down the median of $X$.

Paul is travelling around France to find the perfect baguette. The cost, in euros, of a baguette can be modelled by a continuous random variable $C$ with probability density function $f$ given by

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

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

Explain why the mean cost of a baguette is €7.

Given that the probability that a randomly selected baguette costs less than €9 is $\frac{25}{27}$, find the probability that a randomly selected baguette costs between €5 and €9.

The continuous random variable $X$ has probability density function $f$ given by

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

Find the value of $k$.

Find

(i) $\mathrm{P}(X\le 1)$

(ii) $\mathrm{P}(X\le 1.5)$

(iii) $\mathrm{P}(1\le X\le 1.5)$.

Find

(i) $\mathrm{P}(X\le 1|X\ge 0.5)$

(ii) $\mathrm{P}(X\ge 1.5|X\ge 1)$.

The continuous random variable $X$ has probability density function $f$ given by

$f\left(x\right)=\{\begin{array}{cc}\frac{1}{64}(3x-20),& a\le x<12\\ \frac{1}{64}{\left(x-16\right)}^2,& 12\le x\le 16\\ 0,& \text{otherwise}\end{array}$

where $a$ is a constant.

(i) Given that $\mathrm{P}(12\le X\le 16)=\frac{1}{3}$, find the value of $a$.

(ii) Explain why the median of $X$, $m$, lies in the interval $a\le m\le 12$.

Find

(i) $\mathrm{E}\left(X\right)$

(ii) $\mathrm{E}\left(X^2\right)$

(iii) $\mathrm{Var}\left(X\right)$.

In a quick-fire quiz of 25 questions, contestants have just over 2 seconds to answer each question. The time, in seconds, taken to answer a single question can be modelled by a continuous random variable $T$ with probability density function $f$ given by

$f\left(t\right)=\{\begin{array}{cc}\sin \frac{t}{2},& 0\le t\le T_{\max }\\ 0,& \text{otherwise}\end{array}$

Find the exact value of $T_{\max }$, and verify that it is just over 2 seconds.

Find the probability that a contestant takes between 1 and 2 seconds to answer a question.

Sketch the graph of $y=f\left(t\right)$.

Write down the mode of $T$.

Find the median of $T$.

Estimate the total time that a contestant takes to answer all 25 questions.

The continuous random variable $X$ has probability density function $f$ given by

$f\left(x\right)=\{\begin{array}{cc}\frac{1}{p}\cos (x-q),& a\le x\le a+\pi \\ 0,& \text{otherwise}\end{array}$

Given that $f\left(a\right)=0$, write down, in terms of $a$, the mean, the median and the mode of $X$. Justify your answers.

It is now given that $a=\frac{\pi }{6}$.

(i) Find the smallest positive value of $q$.

(ii) Hence find the value of $p$.

$R$ and $S$ are values of $X$ such that $\mathrm{P}(X<R)=\mathrm{P}(X>S)$. Write down an equation connecting $R$ and $S$.

Two random variables $X$ and $Y$ are such that $\mathrm{E}(3X+5)=\mathrm{Var}(3Y+5)$.

It is also known that $\mathrm{E}\left(Y\right)=\sqrt{\mathrm{E}\left(X\right)}$ and $\mathrm{E}\left(Y^2\right)={\left[\mathrm{E}\left(X\right)\right]}^2$.

Show that $9{\left[\mathrm{E}\left(X\right)\right]}^2-12\mathrm{E}\left(X\right)-5=0$.

Given that $\mathrm{E}\left(X\right)>0$, find $\mathrm{E}\left(X\right)$.

The continuous random variable $X$ has probability density function $f$ given by

$f\left(x\right)=\{\begin{array}{cc}k\mathrm{e}x,& 0\le x\le a\\ k({\mathrm{e}}^2-{\mathrm{e}}^x),& a<x\le 2\\ 0,& \text{otherwise}\end{array}$

where $a$ and $k$ are constants.

Given that $f$ is a continuous function, find the values of $a$ and $k$.

(i) Find $\mathrm{P}(0.5\le X\le 1.5)$.

(ii) Find $\mathrm{P}(X>1)$.

Find the median of $X$.

The continuous random variable $X$ has probability density function $f$ given by

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

where $a$ and $b$ are constants with $a<b$.

(i) Write down an expression for $k$ in terms of $a$ and $b$.

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

(iii) Show that $\mathrm{Var}\left(X\right)=\frac{{\left(b-a\right)}^2}{12}$.

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

The length, in hours, of a company meeting can be modelled by a continuous random variable $T$ with probability density function $f$ given by

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

Find the mean length of a meeting, giving your answer in minutes.

Show that the median of $T$ is greater than the mode of $T$.

The continuous random variable $X$ has probability density function $f$ given by

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

where $a$ and $k$ are constants and $a>2$.

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

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

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

The time, in years, that an investment takes to return a profit can be modelled by a continuous random variable $T$ with probability density function $f$ given by

$f\left(t\right)=\{\begin{array}{cc}\frac{1}{24}\left(a-{\left(t-b\right)}^2\right),& 0\le t\le 3\\ 0,& \text{otherwise}\end{array}$

where $a$ and $b$ are integer constants.

Given that the mode of $T$ is 1, find the values of $a$ and $b$.

Find the expected length of time that a typical investor will have to wait before they make a profit. Give your answer in months, to the nearest month.

A particular company owner will only commit to this investment if there is at least a 75% chance they will make a profit within the first 18 months. Determine whether the company owner will invest or not.

State one limitation of this model.

The continuous random variable $X$ has probability density function $f$ given by

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

Find the value of $k$.

Giving your answers to three significant figures, find

(i) the mean of $X$,

(ii) the mode of $X$.

(i) Write down $\mathrm{P}\left(X=\frac{\pi }{3}\right)$.

(ii) Show that the median, $m$, of $X$ lies in the interval $\frac{\pi }{6}<m<\frac{\pi }{3}$.

The continuous random variable $X$ has probability density function $f$ given by

$f\left(x\right)=\{\begin{array}{cc}\frac{1}{6}(x-1),& 1\le x\le 3\\ \frac{1}{12}(7-x),& 3<x\le 7\\ 0,& \text{otherwise}\end{array}$

Draw a box and whisker diagram for the distribution of $X$. Mark the exact values of the quartiles on your diagram.

The function $f$ is defined by $f\left(x\right)=\frac{1}{54}x(x^2-8x+17)$. The graph of $f$ has a local maximum where $x=p$ and a local minimum where $x=q$.

Sketch the graph of $y=f\left(x\right)$, and state the exact values of $p$ and $q$.

The continuous random variable $X$ has probability density function $g$ given by

$g\left(x\right)=\{\begin{array}{cc}f\left(x\right),& 0\le x\le 6\\ 0,& \text{otherwise}\end{array}$

Show that $g$ is a probability density function.

Write down the mode of $X$.

Find

(i) $\mathrm{P}\left(g\left(X\right)>g\left(p\right)\right)$

(ii) $\mathrm{P}\left(g\left(X\right)<g\left(q\right)\right)$.