Cumulative Distribution Function (Edexcel International A Level (IAL) Maths: Statistics 2): Exam Questions

The continuous random variable, $X$, has a cumulative distribution function $\mathrm{F}\left(x\right)$ given by

$F\left(x\right)=\{\begin{array}{cc}0& x<2\\ \frac{1}{8}{\left(x-2\right)}^3& 2\le x\le 4\\ 1& x>4\end{array}$

Find

(i) $P(X\le 2.5)$

(ii) $P(2.9\le X\le 3.1)$

(iii) $P(X>3)$

(iv) $P(X=4)$.

(i) Find the value of $k$ such that $P(X<k)=0.5$.

(ii) Hence write down the name of the average represented by $k$.

The continuous random variable, $X$, has a cumulative distribution function $\mathrm{F}\left(x\right)$ given by

$F\left(x\right)=\{\begin{array}{cc}0& x<0\\ \frac{3}{4000}{x}^3+\frac{1}{400}x²& 0\le x\le 10\\ 1& x>10\end{array}$

Verify that the median of $X$ lies between $x=7.7$ and $7.8$.

Specify fully the probability density function $\mathrm{f}\left(x\right)$ .

By sketching $\mathrm{f}\left(x\right)$, show that the mode of $X$ is 10.

Show that $E\left(X\right)=\frac{175}{24}.$

Explain why the distribution of $X$ is negatively skewed.

The continuous random variable, $X$, has probability density function $f\left(x\right)$ given by

$f\left(x\right)=\{\begin{array}{l}k{x}^4\\ 0\end{array}\begin{array}{c}-2\le x\le 3\\ \mathrm{otherwise}.\end{array}$

Show that $k=\frac{1}{55}.$

Find, in full, the cumulative distribution function $\mathrm{F}\left(x\right).$

Find the lower quartile of $X$, give your answer to 3 significant figures.

The continuous random variable, $X$, has a cumulative distribution function $\mathrm{F}\left(x\right)$ given by

$F\left(x\right)=\{\begin{array}{cc}0& x<5\\ \frac{x-5}{7}& 5\le x\le 12\\ 1& x\ge 12\end{array}$

Find the probability density function $\mathrm{f}\left(x\right)$ .

Write down the name of the distribution of $X$ .

Find the mean and the variance of $X$.

The continuous random variable $T$ represents the times, in hours, that children at a boarding school take to tidy their rooms. The cumulative distribution function of $T$ is

$F\left(t\right)=\{\begin{array}{cc}0& t<0\\ k\left(9t^2-t^4\right)& 0\le t\le 2\\ 1& t\le 2\end{array}$

Show that $k=\frac{1}{20}$ .

Find the probability that a randomly selected child takes longer than 90 minutes to tidy their room.

Find the probability density function $\mathrm{f}\left(t\right)$ of $T$.

Find the mean time taken for a child to tidy their room.

The mass, in milligrams, of an ant is denoted by the continuous random variable $M$. It is modelled by the probability density function

$\mathrm{f}\left(m\right)=\{\begin{array}{cc}\frac{m-1}{k}& 1\le m<3\\ \frac{1}{k}& 3\le m<5\\ 0& \mathrm{otherwise}.\end{array}$

Show that $k=4$ .

Fully define the cumulative distribution function $\mathrm{F}\left(m\right)$ of $M$.

Find the probability that a randomly selected ant weighs between 2 and 4 milligrams.

The continuous random variable, $X$, has a cumulative distribution function $\mathrm{F}\left(x\right)$ given by

 $\mathrm{F}\left(x\right)=\{\begin{array}{cc}0& x<2\\ a-\frac{b}{x}& 2\le x<4\\ \frac{x^2+17}{c}& 4\le x<7\\ 1& x\ge 7\end{array}$

(i) Explain why $b=2a$.

(ii) Show that $c=66$.

(iii) Show that $4a-b=2$ and hence find the values of $a$ and $b$.

Write down the median of $X$.

Find the lower quartile and the upper quartile of $X$.

Describe the skewness of $X$. Justify your answer.

The continuous random variable, $X$, has probability density function $\mathrm{f}\left(x\right)$ given by

$\mathrm{f}\left(x\right)=\{\begin{array}{cc}\frac{x}{5}& 0<x\le 2\\ \frac{10-2x}{15}& 2<x\le 5\\ 0& \mathrm{otherwise}.\end{array}$

Sketch the probability density function of $X$.

Write down the mode of $X$.

Fully define the cumulative distribution function $\mathrm{F}\left(x\right)$ of $X$.

Find the median of $X$.

Comment on the skewness of the distribution of $X$. Justify your answer.

The continuous random variable, $X$, has a cumulative distribution function $\mathrm{F}\left(x\right)$ given by

$\mathrm{F}\left(x\right)=\{\begin{array}{cc}0& x<a\\ \frac{2x+3}{11}& a\le x\le b\\ 1& x>b\end{array}$

Show that $X$ follows a continuous uniform distribution on the interval $\left[a,b\right]$ and state the values of $a$ and $b$.

Find the mean and the standard deviation of $X$.

Find the interquartile range of $X$.

The continuous random variable, $X$, has a cumulative distribution function $\mathrm{F}\left(x\right)$ given by

$\mathrm{F}\left(x\right)=\{\begin{array}{cc}0& x<3\\ a{x}^2+bx& 3\le x\le 5\\ 1& x>5\end{array}$

Show that $a=\frac{1}{10}$ and find the value of $b$.

Fully define the probability density function $\mathrm{f}\left(x\right)$.

Write down the mode of $X$.

Every morning, Simon jogs for an hour and records his distance, $D$ km. $D$ can be modelled as a continuous random variable with a cumulative distribution function $\mathrm{F}\left(d\right)$ given by

$F\left(d\right)=\{\begin{array}{cc}0& d<8\\ k{\left(d-8\right)}^3& 8\le d<9\\ \frac{55d^2-3600}{19d^2}& 9\le d<10\\ 1& d\ge 10\end{array}$

where $k$ is a constant.

Find the value of $k$.

Find the probability that on a random jog, Simon runs 9.5 km in less than an hour.

Specify fully the probability density function $\mathrm{f}\left(d\right)$ of $D$.

Find the mean distance travelled by Simon on a one-hour jog.

Mrs Wiltshire teaches college students on the Advanced Integration course. Students are given a percentage score, $S\%$,  at the end of the course. The score $S$ can be modelled using a continuous random variable with cumulative distribution function $\mathrm{F}\left(x\right)$ given by

$\mathrm{F}\left(s\right)=\{\begin{array}{cc}0& s<0\\ \frac{3}{10000}{s}^2-\frac{1}{500000}{s}^3& 0\le s\le 100\\ 1& s>100\end{array}$

Show that for $0\le s\le 100$, the probability density function can be written in the form

$\mathrm{f}\left(s\right)=\frac{3}{500000}s(k-s)$

where $k$ is an integer to be found.

Hence write down the mean score.

Find the standard deviation of the scores.

The continuous random variable, $X$, has a cumulative distribution function $\mathrm{F}\left(x\right)$ given by

$\mathrm{F}\left(x\right)=\{\begin{array}{cc}0& x<0\\ k\left(300x+3x^2-x^3-x^4\right)& 0\le x\le 4\\ 1& x>4\end{array}$

Show that $k=\frac{1}{928}.$

Show that the median of $X$ is 1.55 correct to 3 significant figures.

Find the mode of $X$. Fully justify your answer.

Find $\mathrm{E}\left(X\right).$

Comment on the skewness of the distribution of $X$. Give a reason for your answer.

The continuous random variable, $X$, has a cumulative distribution function $\mathrm{F}\left(x\right)$  given by

$\mathrm{F}\left(x\right)=\{\begin{array}{cc}0& x<1\\ a{x}^2+c& 1\le x<2\\ bx+c& 2\le x<5\\ 1& x\ge 5\end{array}$

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

Find the median of $X$.

Find the lower quartile and the upper quartile of $X$.

Describe the skewness of $X$. Give a reason for your answer.

Constance is a motivational speaker who delivers talks to employees at different companies to promote teamwork, her talks last for at least $k$ minutes. The length, in minutes, of a talk made by Constance is modelled by the continuous random variable, $L$, which has a cumulative distribution function given by

$\mathrm{F}\left(L\right)=\{\begin{array}{cc}0& l<k\\ \frac{1}{88}\left(l^2-l-a\right)& k\le l\le 10\\ 1& l>10\end{array}$

where $a$ is a constant.

Find the values of $a$ and $k$.

Find the mean length of time of Constance’s talks.

A company asks Constance to deliver a talk. Find the probability that the talk will last longer than the mean length of time of Constance’s talks.

Without further calculations, state whether the median length of time is smaller than, bigger than or equal to the mean length of time. Give a reason for your answer.

The continuous random variable, $Y$, has probability density function $g\left(x\right)$ given by

$g\left(y\right)=\{\begin{array}{cc}\frac{1}{8}{y}^n& 0\le y\le 2\\ \frac{a}{5y^3}& y>2\\ 0& \mathrm{otherwise}.\end{array}$

where $a$ and $n$ are constants.

The cumulative distribution function $G\left(y\right)$ is given by

$G\left(y\right)=\{\begin{array}{cc}0& y<0\\ r{y}^5& 0\le y\le 2\\ s-\frac{t}{y^m}& y>2\end{array}$

where $r,s,t$ and $m$ are constants.

Find the values of $a,m,n,r,s\mathrm{and}t$.

Glen attends the annual Soaring Sunflowers event where thousands of people gather and compare the sizes of their sunflowers. The continuous random variable $H$ represents the heights, in metres, of the sunflowers at the event. The probability density function of $H$ is given by

$\mathrm{f}\left(h\right)=\{\begin{array}{cc}\frac{16h-4h^2-15}{2}& 1.5\le h<2\\ \frac{5\left(h-2\right)}{3}& 2\le h<3\\ 0& \mathrm{otherwise}.\end{array}$

Specify fully the cumulative distribution function $\mathrm{F}\left(h\right)$  of $H$.

The height of Glen’s sunflower is in the top 10% of all heights.

Find an estimate for the minimum height of Glen’s sunflower. Give your answer to 2 decimal places.

An amusement park has a challenge where guests attempt to remain on a mechanical bull for as long as possible. The times, in seconds, that the guests stay on the mechanical bull can be modelled using a continuous random variable, $T$, with probability density function $\mathrm{f}\left(t\right)$.

$\mathrm{f}\left(t\right)=\{\begin{array}{cc}0& t<0\\ \frac{3}{50}& 0\le t<5\\ \frac{6t-15}{250}& 5\le t<10\\ \frac{1}{t^2}& t\ge 10\end{array}$

Specify fully the cumulative distribution function $\mathrm{F}\left(t\right)$ .

Find the median length of time that a guest is able to stay on the mechanical bull.

Given that a guest has remained on the mechanical bull for 5 seconds, find the probability that the guest will still be on the mechanical bull after another 5 seconds.