This ppt is prepared for engineering class. Random variable and stochastic prosses and probability.

1. 1
Random Variable (r.v): A random variable is a function
whose domain is the set S of all experimental outcomes.
A finite single valued function that maps the set of all
experimental outcomes into the set of real numbers R is
said to be a r.v, if the set is an event for every x
in R.
)(⋅X
S
{ })(| xX ≤ξξ
PILLAI

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4. Conditions for a function to be Random Variable
• Every point in S must correspond to only one value
of the random variable.
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5. Probability Distribution function
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{ }xXPxxF ≤=)(
This is known as probability distribution function of a
random variable.

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9. Probability density function
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Derivative of the distribution function is
known as probability density function (pdf).
.
)(
)(
dx
xdF
xf X
X =

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Additional Properties of a PDF
If for some then
This follows, since implies
is the null set, and for any will be a subset
of the null set.
We have
0)( 0 =xFX ,0x .,0)( 0xxxFX ≤= (3-15)
( ) 0)()( 00 =≤= xXPxFX ξ { }0)( xX ≤ξ
{ })(,0 xXxx ≤≤ ξ
{ } ).(1)( xFxXP X−=>ξ (3-16)
{ } { } ,)()( Ω=>∪≤ xXxX ξξ
PILLAI

13. Example
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17. Solution
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