ECE 302: Lecture 4.3 Cumulative Distribution Function

[Pages:21]ECE 302: Lecture 4.3 Cumulative Distribution Function

Prof Stanley Chan

School of Electrical and Computer Engineering Purdue University

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Outline

Cumulative distribution function (CDF):

FX (x) d=ef P[X x]

(1)

What is a CDF? What are the properties of CDF? How are CDFs related to PDF?

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Definition

Definition

Let X be a continuous random variable with a sample space = R. The cumulative distribution function (CDF) of X is

FX (x) d=ef P[X x].

(2)

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Example

Question. (Uniform random variable) Let X be a continuous random

variable

with

PDF

fX (x)

=

1 b-a

for

a

x

b,

and

is

0

otherwise.

Find

the CDF of X .

Solution. FX (x) =

0,

=

x -a b-a

,

1,

x a, a < x b, x > b.

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Example 2

Question.(Exponential random variable) Let X be a continuous random variable with PDF fX (x) = e-x for x 0, and is 0 otherwise. Find the CDF of X .

Solution.

FX (x) =

0, = 1 - e-x ,

x < 0, x 0.

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Properties 1-3

Theorem Let X be a random variable (either continuous or discrete), then the CDF of X has the following properties: (i) The CDF is a non-decreasing. (ii) The maximum of the CDF is when x = : FX (+) = 1. (iii) The minimum of the CDF is when x = -: FX (-) = 0.

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Property 4

Theorem

Let X be a continuous random variable. If the CDF FX is continuous at any a x b, then

P[a X b] = FX (b) - FX (a).

(3)

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Example

Example 1. (Exponential random variable.) fX (x) = e-x for x 0, FX (x) = 1 - e-x for x 0. Find P[1 X 3].

(a) PDF approach:

P[1 X 3] =

= e-3 - e-

(b) CDF approach: P[1 X 3] =

= e-3 - e-

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