Probability Density Functions
Jake Blanchard
Spring 2010
Uncertainty Analysis for Engineers
1
Random Variables
We will spend the rest of the semester
dealing with random variables
 A random variable is a function defined
on a particular sample space
 For example, if we roll two dice there are
36 possible outcomes – this is the sample
space
 The sum of the two dice is the random
variable

Uncertainty Analysis for Engineers
2
Random Variables
Let y1 and y2 represent the values of the
two dice
 Let x=y1+y2
 x can take on any one of 11 values
between 2 and 12, with some more
common than others
 The relative likelihood of rolling each of
the possible sums is

2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
5
4
3
2
1
Uncertainty Analysis for Engineers
3
Probability Distribution Function
We can calculate a probability from this
table and plot the probability against the
sum
0.18
0.16
Probability of Occurrence

0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
2
3
4
5
6
7
8
Sum of Two Dice
9
10
11
Uncertainty Analysis for Engineers
12
4
Continuous Probability Distribution
Functions

Define the pdf [f(x)]such that the
probability that x falls between a and b is
given by
b
Pr( a  x  b ) 

f ( x ) dx
a
Uncertainty Analysis for Engineers
5
Cumulative Probability
What if we are interested in the
probability that the sum is at or below
some value
 For example, the probability that the sum
is less than or equal to 4 is
6/36=1/6=0.167
 We can plot this value as a function of the
sum

Uncertainty Analysis for Engineers
6
Cumulative Probability
1
0.9
0.8
Cumulative Probability
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
Sum of Two Dice
8
9
10
11
Uncertainty Analysis for Engineers
7
Cumulative Probability
We call this the cumulative distribution
function (CDF)
 It has a minimum of 0, a maximum of 1,
and is monotonic
 For the example of the sum of two dice,
the CDF is

F  xi  

p ( xi )
x  xi
x
or
F ( x) 


f ( z ) dz
Uncertainty Analysis for Engineers
8
Continuous Functions
Consider the decay of a radioactive
particle
 The probability it will survive beyond time
ti is Pr(t>ti)=exp(-ti)
 Hence, the CDF is given by
Pr(t<=ti)=F(ti)=1-exp(-ti)
 This is plotted for =1/s on the next slide

Uncertainty Analysis for Engineers
9
CDF for radioactive decay
1
0.9
0.8
F(time)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
time
Uncertainty Analysis for Engineers
10
Decay Example

For =0.1, the probability that a particle
will decay between 4 and 5 seconds is
given by P(4<t<=5)=F(5)-F(4)=[1-exp(0.5)]-[1-exp(-0.4)]=0.063
Uncertainty Analysis for Engineers
11
Characterizing Distributions
Functions

We will see later how to characterize
these functions using
◦
◦
◦
◦
◦
◦
Mean
Median
Standard Deviation
Skewness
Kurtosis
Etc.
Uncertainty Analysis for Engineers
12
Bivariate Distributions
Sometimes we work with more than one
random variable.
 These can be correlated, so it is
appropriate to define a single pdf that
governs both variables simultaneously
 We call this a joint probability density
function

Uncertainty Analysis for Engineers
13
Joint PDFs

Two continuous random variables are said
to have a bivariate or joint pdf f(x,y) if
Pr  x1  x  x 2 and y 1  y  y 2  
y2 x2

f ( x , y ) dxdy
y 1 x1
f ( x, y )  0
 

f ( x , y ) dxdy  1

F  x1 , y 1  
y 1 x1

f ( x , y ) dxdy

Uncertainty Analysis for Engineers
14
Types of pdfs
We have many choices for functional
forms of pdfs
 Our goal is to represent reality
 Ultimately, we need data to validate our
choice of pdf
 We’ll discuss this later
 Next, we’ll look at some of the common
forms

Uncertainty Analysis for Engineers
15