This document outlines probability density functions (PDFs) including: - The definition of a PDF as describing the relative likelihood of a random variable taking a value. - Properties of PDFs such as being nonnegative and integrating to 1. - Joint PDFs describing the probability of multiple random variables taking values simultaneously. - Marginal PDFs describing probabilities of single variables without reference to others. - An example calculating a joint PDF and its marginals.

2. OUTLINES
• RANDOM VARIABLES
• DEFINITION
• PROPERTIES
• EXAMPLE
• JOINT PDF
• PROPERTIES
• MARGINAL PDF
• EXAMPLE

3. RANDOM VARIABLES
• A random variable has a defined set of values with
different probabilities.
• Random variables
Discrete Continuous
finite number infinite possibilities
of outcomes of values
Eg: Dead/alive, pass/fail, Eg: height, weight,
dice, counts etc speedometer, real numbers etc

4. DEFINITION
• A probability density function (PDF) is a function that
describes the relative likelihood for this random variable to
take on a given value.
• It is given by the integral of the variable’s density over that
range.
• It can be represented by the area under the density function
but above the horizontal axis and between the lowest and
greatest values of the range.

5. PROPERTIES
variable.randomcontinuousa
ofCDFofderivativetheis)(
)(][)(
1)()(
0)()(
Itiv
dxx
b
a
bXaPiii
dxxfii
xfi
X
X
X
f






6. EXAMPLE
• The random variable X has p.d.f given by
10),2()(  xxxkxf
Find k
otherwise,0

7. SOLUTION
The condition is,
1








dxxf
 
1
0
1)2( dxxxk
6k

8. JOINT PDF
The joint PDF for X, Y, ... is a pdf that gives the
probability that each of X,Y, ... falls in any particular
range or discrete set of values specified for that
variable.
The joint PDF is given by
where (X, Y) is a continuous bivariate random
variable.






 dxdyyxfXYyxXYF ),(),(

9. PROPERTIES
 







RA
dxdyyxfXYAYXPiii
dxdyyxfXYii
yxfXYi
),()],[()(
1),()(
0),()(

d
c
b
a
XY dxdyyxfdYcbXaPiv ),(),()(

10. MARGINAL PDF’S
dxyxfxfii
dyyxfxfi
XYy
XYX








),()()(
),()()(
The Marginal PDF gives the probabilities of various values of
the variables in the subset without reference to the values of
the other variables.
The above equations are referred as marginal pdf’s of X and Y.

11. EXAMPLE
Joint PDF is
(1)Find k.
(2)Find marginal PDF.
),( yxfXY







otherwise
xkxy
,0
1y0,10,

12. 1) To find k,
4
1
0
1
0
1
1),(

  






k
dxdykxy
dxdyyxf X Y
SOLUTION

13. yyYf
dxxyyYf
dxkxyyYf
xxXf
dyxyxXf
dykxyx
X
f
2)(
1
0
4)(
1
0
)(
2)(
1
0
4)(
1
0
)(






2) Marginal PDF.