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ELEC 303, Koushanfar, Fall’09 ELEC 303 – Random Signals Lecture 6 – More Discrete Random Variables Farinaz Koushanfar ECE Dept., Rice University Sept 10, 2009

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ELEC 303, Koushanfar, Fall’09 Lecture outline Reading: Section , 2-6 Review Discrete random variables – Examples of PMFs: Binomial, Geometric – Expectation, mean, and variance – Conditioning

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ELEC 303, Koushanfar, Fall’09 Review – random variable A random variable is defined by a deterministic function that maps from the sample space to real numbers

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ELEC 303, Koushanfar, Fall’09 Mean

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ELEC 303, Koushanfar, Fall’09 Mean (Cont’d)

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ELEC 303, Koushanfar, Fall’09 Mean (Cont’d)

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ELEC 303, Koushanfar, Fall’09 Mean (Expectation) Definition: Interpretations – Center of gravity for the PMF – Average in a large number of repetitions for one experiment Figure courtesy of

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ELEC 303, Koushanfar, Fall’09 Mean (Cont’d)

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ELEC 303, Koushanfar, Fall’09 Properties of expectation Let X be a RV and let Y=g(X) – It is often hard to calculate E[Y]= y yP Y (y) – It is easier to compute: E[Y]= x g(x)P X (x) Second moment: E[X 2 ] Generally speaking, E[g(X)] g(E[X]) Variance:

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ELEC 303, Koushanfar, Fall’09 Properties of Expectation If and are constants, and X and Y are RVs: – E[ ]= – E[ X]= – E[ X+ ]= – E[X+Y]= – E[X.Y]=

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ELEC 303, Koushanfar, Fall’09 Variance

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ELEC 303, Koushanfar, Fall’09 Variance (Cont’d)

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ELEC 303, Koushanfar, Fall’09 Variance (Cont’d)

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ELEC 303, Koushanfar, Fall’09 Variance (Cont’d)

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ELEC 303, Koushanfar, Fall’09 Discrete uniform distribution

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ELEC 303, Koushanfar, Fall’09 Average speed vs. average time If weather is good (probability=0.6) Alice walks the 2 miles with speed 5miles/hr. If weather is bad, Alice rides her motorcycle at a speed V=30 miles/hr. What is the mean of the time T to get to the class? Correct Solution - derive the PMF of T: P T (t)=0.6, if t=2/5; P T (t)=0.4, if t=2/30 E[T] = 0.6 2/ 2/30 = 4/15 hrs = 16 mins

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ELEC 303, Koushanfar, Fall’09 Average speed vs. average time If weather is good (probability=0.6) Alice walks the 2 miles with speed 5miles/hr. If weather is bad, Alice rides her motorcycle at a speed V=30 miles/hr. What is the mean of the time T to get to the class? Mistake: it is wrong to find the ave speed E[V] = 0.6 30 = 15 miles/hr E[T] = 2/E[V] = 2/15 hrs = 8 mins Summary: E[T] = E[2/V] 2/E[V]

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ELEC 303, Koushanfar, Fall’09 Review: discrete random variable PMF, expectation, variance Probability mass function (PMF) P X (x) = P (X=x) x P X (x)=1

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ELEC 303, Koushanfar, Fall’09 Some properties of expectation

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ELEC 303, Koushanfar, Fall’09 Bernoulli (indicator) RV

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ELEC 303, Koushanfar, Fall’09 Binomial RV

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ELEC 303, Koushanfar, Fall’09 Conditional PMF and expectation

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ELEC 303, Koushanfar, Fall’09 Geometric PMF

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ELEC 303, Koushanfar, Fall’09 Geometric PMF

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ELEC 303, Koushanfar, Fall’09 Total expectation theorem

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ELEC 303, Koushanfar, Fall’09 Geometric random variable

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ELEC 303, Koushanfar, Fall’09 Geometric random variable

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