probability density function (pdf)
TRANSCRIPT
Probability Density Function (PDF)
OUTLINES
• RANDOM VARIABLES• DEFINITION• PROPERTIES• EXAMPLE• JOINT PDF• PROPERTIES• MARGINAL PDF• EXAMPLE
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
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.
PROPERTIES
variable.random continuous a
of CDF of derivative theis )(
)(][)(
1)()(
0)()(
Itiv
dxxb
abXaPiii
dxxfii
xfi
X
X
X
f
EXAMPLE
• The random variable X has p.d.f given by
10 ),2()( xxxkxf
Find k
otherwise ,0
SOLUTION
The condition is,
1
dxxf
1
01)2( dxxxk
6k
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 ),(),(
PROPERTIES
RAdxdyyxfXYAYXPiii
dxdyyxfXYii
yxfXYi
),()],[()(
1 ),()(
0 ),()(
d
c
b
a
XY dxdyyxfdYcbXaPiv ),(),()(
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.
EXAMPLE
Joint PDF is
(1)Find k.
(2)Find marginal PDF.
),( yxf XY
otherwise
xkxy ,0
1y0 ,10 ,
1) To find k,
4
1
0
1
01
1),(
k
dxdykxy
dxdyyxf XY
SOLUTION
yyYf
dxxyyYf
dxkxyyYf
xxXf
dyxyxXf
dykxyxX
f
2)(
1
0
4)(
1
0
)(
2)(
1
0
4)(
1
0
)(
2) Marginal PDF.
THANK YOU