The continuous random variable, , has a cumulative distribution function given by
Find:
Write the following in terms of and .
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The continuous random variable, , has a cumulative distribution function given by
Find:
Write the following in terms of and .
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The continuous random variable, , has probability density function given by
The cumulative distribution function is given by
where is a function and are constants.
Explain why and write down the value of .
Use integration to show that
where are constants to be found and is a constant of integration.
Use to find the value of . Verify this by checking that .
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The continuous random variable, , has a cumulative distribution function given by
The probability density function is given by
where is a function and is a constant.
Write down the value of .
Use differentiation to find .
Sketch a graph of .
Use the graph to write down the mode of
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The continuous random variable is uniformly distributed over the interval .
Find:
Specify fully the probability density function
Specify fully the cumulative distribution function
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The continuous random variable, , has probability density function given by
The cumulative distribution function is given by
where and are functions.
Explain why
Use integration and part (a) to find and .
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The continuous random variable, , has a cumulative distribution function given by
The median of is denoted by .
Using differentiation, find the probability density function, , of the random variable for all values of .
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The continuous random variable, , has a cumulative distribution function given by
Find
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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.
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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.
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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 .
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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.
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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.
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The continuous random variable, , has a cumulative distribution function given by
Write down the median of .
Find the lower quartile and the upper quartile of .
Describe the skewness of . Justify your answer.
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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.
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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 .
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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 .
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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.
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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.
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The continuous random variable, , has a cumulative distribution function given by
Show that
Show that the median of is 1.55 correct to 3 significant figures.
Find the mode of . Fully justify your answer.
Find
Comment on the skewness of the distribution of . Give a reason for your answer.
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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.
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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.
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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 .
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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.
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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.
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