Lecture10_Spring11

Lecture10_Spring11 - Elec 210: Lecture 10 Single random...

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Elec210 Lecture 10 1 Elec 210: Lecture 10 Single random variables: discrete, continuous and mixed Continuous R.V. and Cumulative Distribution Function (CDF) Probability Density Function (PDF) Conditional CDF’s and PDF’s
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Elec210 Lecture 10 2 Random Variables: Review A random variable X is a function that assigns a number to every outcome of an experiment. The function is fixed and deterministic. All randomness in the observed value is due to the underlying experiment ξ 1 IR X ξ () x = S x 1 ξ 2 x 2 Range S X Domain
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Elec210 Lecture 10 3 Specifying Probabilities of Random Variables So far, we have focused on discrete random variables , Possible values come from a countable or finite set. Probability of any event can be computed from the probability mass function . But, in many cases, possible values are not naturally restricted to a countable or finite set, but rather come from an uncountably infinite set e.g., the unit interval or the real line. For random variables like this, we cannot use the probability mass function , but we must use one of the following: 1. Cumulative distribution function 2. Probability density function 3. Characteristic function
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Elec210 Lecture 10 4 Cumulative Distribution Function (CDF) A random variable X is a function from the sample space S to R with the property that the set is in the event set for every b in R Cdf defined as: Relative frequency interpretation: if the experiment is performed a large number of times, F X ( x ) is the proportion of times that the value of X ended up less than or equal to x . Although F X ( x ) gives the probabilities of only one type of event (semi- infinite intervals), the probability of any event of interest can be computed from it using the axioms and corollaries . < < = x x X P x F X - for ] [ ) ( x X S () {} b X A b = ζ :
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5 Example 4.1: Three Coin Tosses Let X be the number of heads observed in three tosses of a fair coin. This is a discrete random variable that assumes values in S X = {0,1,2,3} Properties of this cdf: Piecewise constant Can be expressed as sum of unit steps Continuous from the right Consider the discontinuity at 1. The limit from the left is 1/8. •F o r δ a small positive number, The limit from the right is 1/2. CDF value at 1 is the limit from the right!! cdf 33 11 8888 0123 X X p 1 8 (1 ) [ 1 ] {0 heads} X FP X P δδ −= = 1 2 ) [ 1 ] {0 or 1 heads} X X P += = 1 2 (1) [ 1] {0 or 1 heads} X X P =≤ = = Elec210 Lecture 10
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Elec210 Lecture 10 6 Unit Step Function The unit step function is Useful to denote functions that are zero for negative values: Example: We can build piecewise constant functions as sums of scaled and shifted steps: < = 0 if 0 0 if 1 ) ( x x x u x u ( x ) 1 () < = 0 if 0 0 if 1 ) ( 1 x x e x u e x x λ x a ) ( 0 x x u a ) ( 0 x x u a 0 x
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Elec210 Lecture 10 7 CDF = Sum of Step Functions () 3 8 1 2 8 3 1 8 3 8 1 + + + = x u x u x u x u x F X x u 8 1 1 8 3 x u 2 8 3 x u 3 8 1 x u x x x x x x F X
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Elec210 Lecture 10 8 Example:
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This note was uploaded on 03/27/2011 for the course ELEC 202 taught by Professor ? during the Spring '11 term at HKUST.

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Lecture10_Spring11 - Elec 210: Lecture 10 Single random...

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