1.4 Cumulative Distribution Function

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

Sketch the probability density function of .

Write down the mode of .

Fully define the cumulative distribution function  of .

Find the median of .

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

The continuous random variable, , has a cumulative distribution function  given by

Show that  follows a continuous uniform distribution on the interval  and state the values of  and .

Find the mean and the standard deviation of .

Find the interquartile range of .

The continuous random variable, , has a cumulative distribution function  given by

Show that  and find the value of .

Fully define the probability density function .

Write down the mode of .

Every morning, Simon jogs for an hour and records his distance,  km.  can be modelled as a continuous random variable with a cumulative distribution function  given by

where  is a constant.

Find the value of .

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

Specify fully the probability density function  of .

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, ,  at the end of the course. The score  can be modelled using a continuous random variable with cumulative distribution function  given by

Show that for , the probability density function can be written in the form

where  is an integer to be found.

Hence write down the mean score.

Find the standard deviation of the scores.

The continuous random variable, , has a cumulative distribution function  given by

Verify that the median of  lies between and .

Specify fully the probability density function .

By sketching , show that the mode of  is 10.

Show that 

Explain why the distribution of is negatively skewed.

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

Show that 

Find, in full, the cumulative distribution function 

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

The continuous random variable, , has a cumulative distribution function  given by

Find the probability density function .

Write down the name of the distribution of .

Find the mean and the variance of .

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

Show that .

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

Find the probability density function  of .

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 . It is modelled by the probability density function

Show that .

Fully define the cumulative distribution function  of .

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

The continuous random variable, , has a cumulative distribution function  given by

Find the values of and .

Find the median of .

Find the lower quartile and the upper quartile of .

Describe the skewness of . 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  minutes. The length, in minutes, of a talk made by Constance is modelled by the continuous random variable, , which has a cumulative distribution function given by

where  is a constant.

Find the values of  and .

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, , has probability density function  given by

where  and  are constants.

The cumulative distribution function  is given by

where and  are constants.

Find the values of .

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

Specify fully the cumulative distribution function   of .

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, , with probability density function .

Specify fully the cumulative distribution function .

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.