Further Probability Distributions (DP IB Maths: AA HL)

A continuous random variable  has the probability density function given by

Find the value of .

Giving your answers to three significant figures, find

(i)

the mean of ,

(ii)

the mode of .

The continuous random variable  has probability density function

Find the value of .

Sketch the probability density function.

Find: 

(i)

(ii)

(iii)

.

The discrete random variable,  , has probability distribution function

Show that .

Find the expected value and variance of .

As part of a game, a four-sided spinner is created with the numbers 2, 4, 6 and 12. The discrete random variable  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,  kWh, follows the following probability density function

Show that .

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

Consider the function  defined by,

Use your GDC to verify that can represent a probability density function for a continuous random variable  in the case when .  Explain your verification process in full.

Use your GDC to find the mode of .

Use your GDC to estimate the median of .

Consider the probability density function for a continuous random variable, 

Given that , find the value of .

Briefly explain how, without further calculations, it can be deduced that the median of , , lies in the interval

Find

(i)

(ii)

(iii)

. 

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, , which has probability density function

Find the exact value of 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

Write down the mode of .

Find the median of .

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

The continuous random variable, , has probability density function

Given that , write down, in terms of , the mean, median and mode of , briefly explaining how you obtained your answers.

and  are values of such that .
Write down an equation connecting  and .

Two random variables,  and  are such that .

It is also known that and .

Show that  .

Given that , find .

Consider the function

Sketch the graph of .

Write down a domain for such that it could be a probability density function for a continuous random variable .

Write down

(i)

,

(ii)

,

(iii)

The median of .

The continuous random variable  has the probability density function

where and  are constants.

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

Find the median of .

The continuous random variable, , follows a uniform distribution.

The probability density function is given by

Given that and , find the values of  and .

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

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  has probability density function

Show that .

Given that  show that   and hence find the value of .

Find the exact value of .

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

Sketch the graph of .

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

Given that the probability that a randomly selected baguette costs less than €9 is , find the probability that a randomly selected baguette costs between €5 and €9

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

where  and  are integer constants.

Given that the mode of is 1, find the values of  and .

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, , has probability density function given by

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

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

	1	2	3	4	5	6	7	8	9

where .

Show that .

Find

(i)

(ii)

.

Find

(i)

(ii)

, where  is a constant.

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

Given that , find the value of .

Find

(i)

(ii)

(iii)

Find

(i)

(ii)

The function defined by  has a local maximum when  and a local minimum when .

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

The continuous random variable, , has probability density function

Briefly explain how it can be verified that is a suitable model for a probability density function.

Write down the mode of .

Find the probability that

(i)

is such that

(ii)

is such that  .