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Probability And Random Variable Lecture(5)

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

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Probability And Random Variable Lecture(5)

  1. 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. 4. Conditions for a function to be Random Variable • Every point in S must correspond to only one value of the random variable. 4
  5. 5. Probability Distribution function 5 { }xXPxxF ≤=)( This is known as probability distribution function of a random variable.
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  9. 9. Probability density function 9 Derivative of the distribution function is known as probability density function (pdf). . )( )( dx xdF xf X X =
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  12. 12. 12 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. 13. Example 13
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  17. 17. Solution 17
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