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How many words can you make?

Three or more lettersMust all include A

A

PI

Y

M

LF

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The Rectangular Distribution

A special continuous distributionAlso called “The Uniform Distribution”

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0.2

The rectangular distribution (or uniform distribution)

A continuous random variable has the distribution:

otherwise0

61)(

xkxf

1 6 x

f(x)

area = 1Find k

Sketch the distribution

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The rectangular distribution (or uniform distribution)

A continuous random variable has the distribution:

otherwise0

61)(

xkxf

1 6 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 = 1Find k

Sketch the distribution0.2

We can also write X ~ R(1, 6)

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Spot the pattern…

Here are some more rectangular distributions. What do the ranges of x all have in common with their

pdfs?

1 6

51

x

f(x)

8 12

41

x

f(x)

99 101

21

x

f(x)

a b x

f(x)

?

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Definition of a rectangular (uniform) distribution.

A continuous r.v. X having pdf f(x) where

We say that X ~ R (a, b)

bxaforab

xf

1

)(

a b

ab 1

x

f(x)

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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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Expectation and Variance

Uniform Distribution

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

otherwise0

612.0)(

xxf

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Important Results

E(X)

Var(X)

2ba

So, E(X) = mean = median

12

2ab

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a b x

f(x)

ab 1

Show that(a) E(X)

(b) Var(X)

2ba

12

2ab

Proofs to learn

For ANY Rectangular Distribution

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Finding E(X) and Var(X)

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Example 2

0.57

3

3

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Example 3

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Example 4