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Probability Density Function (PDF)

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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.

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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.randomcontinuousa ofCDFofderivativetheis)( )(][)( 1)()( 0)()( Itiv dxx b a bXaPiii 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 0 1)2( dxxxk 6k
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          RA dxdyyxfXYAYXPiii 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. ),( yxfXY        otherwise xkxy ,0 1y0,10,
1) To find k, 4 1 0 1 0 1 1),(           k dxdykxy dxdyyxf X Y SOLUTION
yyYf dxxyyYf dxkxyyYf xxXf dyxyxXf dykxyx X f 2)( 1 0 4)( 1 0 )( 2)( 1 0 4)( 1 0 )(       2) Marginal PDF.
Probability Density Function (PDF)

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