# chap2-slide - Probability and Stochastic Processes A...

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Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers Chapter 2 Viewgraphs 1

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Random Variables (RV) Experiment: Procedure + Observations Observation is an outcome Assign a number to each outcome: Random variable 2
Random Variables Three ways to get a RV: The RV is the observation The RV is a function of the observation The RV is a function of a RV 3

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Discrete Random Variables S X = range of X (set of possible values) X is discrete is S X is countable Discrete RV X has PMF P X ( x ) = P [ X = x ] 4
PMF Properties P X ( x ) 0 x S X P X ( x ) = 1 For an event B S X , P [ B ] = P [ X B ] = X x B P X ( x ) 5

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Bernoulli RV Get the phone number of a random student. Let X = 0 if the last digit is even. Otherwise, let X = 1 P X ( x ) = 1 - p x = 0 p x = 1 0 otherwise 6
Binomial RV Test n circuits, each circuit is rejected with probability p independent of other tests K = number of rejects K is the number of successes in n trials: P K ( k ) = ( n k ) p k (1 - p ) n - k k = 0 , 1 ,...,n 0 otherwise 7

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Circuit rejected with prob p . Y is the number of tests up to and including the ﬁrst reject ' ' ' ' ' r p a 1 - p ' ' ' ' ' r p a 1 - p ' ' ' ' ' r p a 1 - p Y =1 Y =2 Y =3 ... From the tree,
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chap2-slide - Probability and Stochastic Processes A...

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