How many words can you make?

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

2. The Rectangular Distribution A special continuous distribution Also called “The Uniform Distribution”

3. f(x) x 1 6 The rectangular distribution (or uniform distribution) • A continuous random variable has the distribution: Find k Sketch the distribution area = 1 0.2

4. f(x) x 1 6 The rectangular distribution (or uniform distribution) • A continuous random variable has the distribution: Find k Sketch the distribution area = 1 0.2 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. We can also write X ~ R(1, 6)

5. f(x) f(x) f(x) x x 8 12 1 6 f(x) 99 101 x ? x a b Spot the pattern… • Here are some more rectangular distributions. • What do the ranges of x all have in common with their pdfs?

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

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

9. Expectation and Variance Uniform Distribution

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

11. Important Results • E(X) • Var(X) So, E(X) = mean = median

12. Show that • E(X) • Var(X) f(x) x a b Proofs to learn For ANY Rectangular Distribution

13. Finding E(X) and Var(X)

15. Example 2 0.57 3 3

16. Example 3

17. Example 4