1. The Discrete Uniform Distribution © Christine Crisp “Teach A Level Maths” Statistics 1

2. The Discrete Uniform Distribution "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Statistics 1 Edexcel

3. The Discrete Uniform Distribution We may have a situation where the probabilities of each event are the same. If X is the random variable (r.v.) “ the number showing”, the probability distribution table is P(X = x) 654321x The probability distribution function (p.d.f.) is For example, if we roll a fair die, we assume that the probability of obtaining each number is. The distribution with equal probabilities is called “uniform”

4. The Discrete Uniform Distribution It is also possible to have a continuous uniform distribution where the r.v. can be any number in a given interval. The continuous uniform distribution is also called the rectangular distribution. You will not be studying it in this module.

5. The Discrete Uniform Distribution The mean value of X is given by the average of the 1 st and last values of x, so, A diagram for the distribution looks like this: p x 1 2 34 5 6

6. The Discrete Uniform Distribution However, we could also use the formula for the mean of any discrete distribution of a random variable: For we would get

7. The Discrete Uniform Distribution Solution: (a) The sum of the probabilities, e.g. 1 The r.v. X has p.d.f. given by where k is constant. (a) Find the value of k. (b) Find the mean of the distribution. (b) The mean, , is given by Either Or Be careful here! The probabilities are the same for all the values of x.

8. The Discrete Uniform Distribution We can find the variance for any discrete random variable X using The Variance of the Uniform Distribution e.g. The random variable X has p.d.f. given by So, We found earlier that   3·5, so

9. The Discrete Uniform Distribution The formulae for the mean and variance of the discrete uniform distribution are not in the formulae booklet. However, the mean can either be seen directly from the x -values or by using the general formula for the mean of a discrete distribution. However, if you forget the formula, you can use the general formula for variance as I just did in the example. A formula for the variance is given by where n is the number of values of X.

10. The Discrete Uniform Distribution SUMMARY  The mean of the distribution can be found by averaging the 1 st and last x -values  If a random variable X has p.d.f. given by where k is constant and the values of x are discrete, then X has a discrete uniform distribution.  The value of k is found by using  The variance is given by where n is the number of values of X. The mean and variance can both be found using the formulae for general discrete distributions.

11. The Discrete Uniform Distribution Exercise Solution: 1. The r.v. X has p.d.f. given by where k is constant. (a) Find the value of k. (b) Find the mean and variance of the distribution. (a) (b) We can get the mean directly from the x values or by using the formula. or

12. The Discrete Uniform Distribution If you use this formula, be careful. n is the number of values of x, which in this example is 5, not the final value, 8. where k is constant. (a) Find the value of k. (b) Find the mean and variance of the distribution. 1. The r.v. X has p.d.f. given by Either: Or:

14. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.

15. The Discrete Uniform Distribution SUMMARY  The mean of the distribution can be found by averaging the 1 st and last x -values  If a random variable X has p.d.f. given by where k is constant and the values of x are discrete, then X has a discrete uniform distribution.  The value of k is found by using  The variance is given by where n is the number of values of X. The mean and variance can both be found using the formulae for general discrete distributions.

16. The Discrete Uniform Distribution Solution: (a) The sum of the probabilities, e.g. 1 The r.v. X has p.d.f. given by where k is constant. (a) Find the value of k. (b) Find the mean and variance of the distribution. (b) The mean, , is given by Either Be careful here! The probabilities are the same for all the values of x.

17. The Discrete Uniform Distribution and   2·5, so The variance is given by Or Either Or