1. 22/10/2015 How many words can you make? Three or more letters Must all include A A P I Y M L F

2. 22/10/2015 The Rectangular Distribution A special continuous distribution Also called “The Uniform Distribution”

3. 22/10/2015 0.2 The rectangular distribution (or uniform distribution) A continuous random variable has the distribution: 16 x f(x) area = 1 Find k Sketch the distribution

4. 22/10/2015 The rectangular distribution (or uniform distribution) A continuous random variable has the distribution: 16 x f(x) The distribution is said to be “uniform” (or rectangular) because each value of x in the given range of values has the same probability of occurring. area = 1 Find k Sketch the distribution 0.2 We can also write X ~ R(1, 6)

5. 22/10/2015 Spot the pattern… Here are some more rectangular distributions. What do the ranges of x all have in common with their pdfs? 16 x f(x) 812 x f(x) 99101x f(x) ab x ?

6. 22/10/2015 Definition of a rectangular (uniform) distribution. A continuous r.v. X having pdf f(x) where We say that X ~ R (a, b) ab x f(x)

7. 22/10/2015 Example 1 – Finding Probabilities X is distributed uniformly, where 5 ≤ x ≤ 12 Find P(7 ≤ x ≤ 10) Always visualise it!! x f(x)

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9. Expectation and Variance Uniform Distribution

10. 22/10/2015 Find E(X) and Var(X) from first principles

11. 22/10/2015 Important Results E(X) Var(X) So, E(X) = mean = median

12. 22/10/2015 ab x f(x) Show that (a)E(X) (b)Var(X) Proofs to learn For ANY Rectangular Distribution

13. 22/10/2015 Finding E(X) and Var(X)

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15. Example 2 0.57 3 3

16. 22/10/2015 Example 3

17. 22/10/2015 Example 4