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

A continuous random variable $X$ has the probability density function given by

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

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 $\mathrm{P}\left(X=\frac{\mathrm{\pi }}{3}\right)$

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

The continuous random variable $X$ has probability density function

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

Find the value of $k$.

Sketch the probability density function.

Find: 

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

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

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

The discrete random variable,  $X$, has probability distribution function

$f\left(x\right)=\{\begin{array}{ll}\frac{k}{2x},& x=2,4,6,12\\ 0,& \mathrm{otherwise}\end{array}$

Show that $k=2$.

Find the expected value and variance of $X$.

As part of a game, a four-sided spinner is created with the numbers 2, 4, 6 and 12. The discrete random variable $X$ is used to model the number that the spinner lands on. The score allocated to a player on their turn is 4 more than double the value the spinner lands on.

Find the expected value and variance of a player’s score from a single spin.

A UK energy company charges £0.22 per kilowatt hour (kWh) of electricity used.
The amount of energy used per day by the company’s customers, $X$ kWh, follows the following probability density function

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

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,

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

Use your GDC to verify that $f\left(x\right)$can represent a probability density function for a continuous random variable $X$ in the case when $k=12$.  Explain your verification process in full.

Use your GDC to find the mode of $X$.

Use your GDC to estimate the median of $X$.

Consider the probability density function for a continuous random variable, $X$

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

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

(ii) Briefly explain how, without further calculations, it can be deduced that 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 consisting of 25 questions, contestants have just over 2 seconds to answer each question. The time taken on any single question in the quiz is modelled by the continuous random variable, $T$, which has probability density function

$f\left(t\right)=\{\begin{array}{ll}\sin \frac{1}{2}t,& 0\le t\le T_{\text{max}}\\ 0,& \mathrm{otherwise}\end{array}$

Find the exact value of $T_{\max }$ and verify that this 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 $T$.

Find the mean time for a contestant to answer all 25 questions.

The continuous random variable, $\theta$, has probability density function

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

Given that $f\left(a\right)=0$, write down, in terms of $a$ , the mean, median and mode of $\theta$, briefly explaining how you obtained your answers.

(i) Given that $a=\frac{\mathrm{\pi }}{6}$, deduce the smallest positive value of $q$

(ii) Hence, or otherwise, find the value of $p$

$R$ and $S$ are values of $\theta$ such that $\mathrm{P}(\theta <R)=\mathrm{P}(\theta >S)$. 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 $\mathrm{E}\left(X\right)$.

Consider the function

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

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

Write down a domain for $f\left(x\right)$ such that it could be a probability density function for a continuous random variable $X$.

Write down

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

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

(iii) The median of $X$.

The continuous random variable $X$ has the probability density function

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

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, $X$, follows a uniform distribution.

The probability density function is given by

$f\left(x\right)=\{\begin{array}{ll}k& a\le x\le b\\ 0& \mathrm{otherwise}.\end{array}$

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

(ii) Write down an expression for $\mathrm{E}\left(X\right)$ 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 $b$.

A company meeting lasts $T$ hours. The continuous random variable $T$ can be modelled by the probability density function

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

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

Show that the median meeting time is greater than the modal meeting time. Fully explain your solution.

The continuous random variable $X$ has probability density function

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

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

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

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

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 amount of time, $t$ years, it takes for an investment to return a profit is modelled by a continuous random variable, $T$, with probability density function

$f\left(t\right)=\{\begin{array}{ll}\frac{1}{24}\left(a-{\left(t-b\right)}^2\right)& 0<t<3\\ 0& \mathrm{otherwise}\end{array}$

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.

Give a criticism of the model.

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

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

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

The discrete random variable, $X$ , has the probability distribution given in the table below

where $k\in {\mathrm{\mathbb{Z}}}^{+}$ .

Show that $k=3$ .

Find

(i) $\mathrm{P}(X>4|X\le 7)$

(ii) $\mathrm{P}\left(X\mathrm{is}\mathrm{prime}\right|X\le 5)$.

Find

(i) $\mathrm{E}(4X+5)$

(ii) $\mathrm{Var}(6X+q)$, where $q$ is a constant.

A continuous random variable, $X$, has a probability distribution such that

$P(X\le x)=k{x}^3,0\le x\le 2$

Given that $\mathrm{P}(X\le 2)=1$ , 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 function defined by $f\left(x\right)=\frac{1}{54}x\left(x^2-8x+17\right)$ has a local maximum when $x=p$ and a local minimum when $x=q$.

Sketch the graph of $f$ . State the values of p and q.                                 

The continuous random variable, $X$, has probability density function

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

Briefly explain how it can be verified that $g\left(x\right)$ is a suitable model for a probability density function.

Write down the mode of $X$.

Find the probability that

(i) $X$ is such that $g\left(x\right)>g\left(p\right),$

(ii) $X$ is such that  $g\left(x\right)<g\left(q\right)$.