Cumulative Distribution Functions and Expected Values

10/3/11

Cumulative Distribution Functions and Expected Values

MATH 3342 SECTION 4.2

The Cumulative Distribution Function (cdf)

The cumulative distribution function F(x) for a continuous RV X is defined for every number x by:

x

F(x) = P(X x) = f (y)dy -

For each x, F(x) is the area under the density curve to the left of x.

1

The Uniform Distribution

Recall:

A continuous RV X is said to have a uniform distribution over the interval [A, B] if the pdf is:

# %

1

f (x; A, B) = $ B - A

% &

0

'

AxB % (

otherwise

% )

10/3/11

The Uniform cdf

The cdf of the uniform distribution is obtained as

follows:

F(x) =

x

f (y)dy =

x

1

dy

-

A B-A

=

B

1 -

A

[

y

]

x A

= x-A B-A

2

The Uniform cdf

More completely:

# %

0

% F(x) = $

x-A

% B-A

&% 1

x ................
................

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