Todays Schedule

1
Todays Schedule

• Reading Lathi 10.2
• Quiz 1
• Mini-Lecture 1
• Random Variables
• Probability Density Functions
• Cumulative Distribution Functions
• Activity

2
Axioms of Probability

• Probability theory is based on 3axioms
• P(A) gt0
• P(S) 1
• P(AB) P(A) P(B) IFF P(AB) f

3
Random Variables

• Defn A real-valued random variable (RV) is a
  real-valued function defined on the events of the
  probability system

4
Cumulative Density Function

• The cumulative density function (CDF) of the RV,
  x, is given by Fx(a)Px(xlta)

Fx(a)
1
0.5
a
3
-2
-1
0
5
Probability Density Function

• The probability density function(PDF) of the RV x
  is given by f(x)
• Shows how probability is distributed across the
  axis

6
Types of Distributions

• Discrete-M discrete values at x1, x2, x3,. . . ,
  xm
• Continuous- Can take on any value in an defined
  interval

fx(x)
Fx(a)
1
1
0.5
0.5
x
x
1
-1
1
-1
0
0
7
CDF Properties

• Fx(a) is a nondecreasing function
• 0 lt Fx(a) lt 1
• Fx(-infinity) 0
• Fx(infinity) 1
• F(a) is right-hand continuous

8
PDF Properties

• fx(x) is nonnegative, fx(x) gt 0
• The total probability adds up to one

Fx(a)
2
1
fx(x)
1
-1
1
-1
0
9
Calculating Probability

• To calculate the probability for a range of values

F(b)
2
fx(x)
F(a)
b
a
1
-1
b
a
1
-1
0
10
Example Triangle Wave

• Calculate Probability that the amplitude of a
  triangle wave is greater than 1 v, if A2.

fx(x)
s(t)
A
fx(x)
A
1/2A
-A
-A
-A
0
A
-1
11
Example Triangle Wave

• Calculate Probability that the amplitude of a
  triangle wave is greater than 1 v, if A2.

1
FV(v)
fV(v)
3/4
1/4
-2
2
0
1
-2
0
1
2
12
Example Triangle Wave

• Calculate Probability that the amplitude of a
  triangle wave is in the range 0.5,1 v, if A2.

1
FV(v)
fV(v)
3/4
5/8
1/4
-2
2
0
1
-2
0
1
2
13
Activity Square Wave

• Calculate Probability that the amplitude of a
  square wave is at A.
• Sketch PDF and CDF

fx(x)
s(t)
A
fx(x)
A
-A
-A
-A
0
A
-1
14
Next Time

• Reading Lathi 10.2-10.3
• Mini-Lecture 1
• Examples of PDFs and CDFs
• Activity
• Mini-Lecture 2
• Time Averages and Moments
• Activity
• Mini-Lecture 3
• More Distributions and Moments

15
Activity Square Wave

• Calculate Probability that the amplitude of a
  square wave is at A. 1/4
• Sketch PDF and CDF

fx(x)
s(t)
A
Fx(x)
A
-A
0
-A
A
-A
0