# Ω x ω a a 1 3 5 a 1 a 1 so x is not a random

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ω X (ω) a a < 0 1 , 3 , 5 0 a < 1 a 1 So X is not a random variable — it not measurable with respect to F . The field F is too “coarse” to measure if ω is odd. On the other hand, let us now define X (ω) 0 ω 3 1 ω > 3 Is this a random variable? 2 We observethat a randomvariable cannotgenerate partitions of the underlyingsample space which are not events in the σ -field F . Another way of saying that X is measurable: For any B B , ω : X (ω) B F . Recalling the idea of Borel sets B associated with the real line, we see that a random variable is a measurable function from ( , F ) to ( R , B ) : X : ( , F ) ( R , B ). We will abbreviate the term random variable as r.v. . We will use a notational shorthand for random variables. Definition 8 Suppose ( , F , P ) is a probability space and X : ( , F ) ( R , B ) . For each B B we define P ( X B ) P ( ω : X (ω) B ). 2 By this definition, we can identify a new probability space. Using ( R , B ) as the pre- probability space, we use the measure P X ( B ) P ( X B ). So we get the probability space ( R , B , P X ) . As a matter of practicality, if the sample space is R , with the Borel field, most mappings to R will be random variables. To summarize: ( , F , P ) X ( R , B , P X ) where P X ( B ) P ( ω X (ω) B ) for B B . Distribution functions The cumulative distribution function (cdf) of an r.v. X is defined for each a R as F X ( a ) P ( X a ) P ( ω X (ω) a ) P X (( , a ] ) Properties of cdf:

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ECE 6010: Lecture 1 – Introduction; Review of Random Variables 7 1. F X is non-decreasing: If a < b then F X ( a ) F X ( b ) . 2. lim a F X ( a ) 1 3. lim a F X ( a ) 0 . 4. F X is right-continuous: lim b a F X ( b ) F X ( a ) . Draw “typical” picture. These four properties completely characterize the family of cdfs on the real line. Any function which satisfies these has a corresponding probability distribution. 5. For b > a : P ( a < X b ) F X ( b ) F X ( a ) . 6. P ( X a 0 ) F X ( a 0 ) lim a a 0 F x ( a ) . Thus, if F X is continuous at a 0 , P ( X a 0 ) 0. From these properties, we can assign probabilities to all intervals from knowledge of the cdf. Thus we can extend this to all Borel sets. Thus F X determines a unique probability distribution on ( R , B ) , so F X and P X are uniquely related. Pure types of r.v.s 1. Discrete r.v.s — an r.v. whose possible values can be enumerated. 2. Continuous r.v.s — an r.v. whose distribution function can be written as the (regular) integral of another function 3. Singular, but not discrete — Any other r.v. Discrete r.v.s A random variable that can take on at most a countable number of possible values is said to be a discrete r.v.: X : x 1 , x 2 , . . . Definition 9 For a discrete r.v. X , we define the probability mass function (pmf) (or discrete density function) by p X ( a ) P ( X a ) a R where p X ( a ) 0 if a x i for any r.v. outcome x i . 2 Properties of pmfs: 1. Nonnegativity: p X ( a ) 0 a x 1 , x 2 , . . . 0 else
ECE 6010: Lecture 1 – Introduction; Review of Random Variables 8 2. Total probability: i 1 p X ( x i ) 1 . (These two properties completely characterize the class of all pdfs on a given set x 1 , x 2 , . . . .) 3. Relation to cdf: p X ( a ) F X ( a ) lim b a F X ( b ) (To the pdf and the cdf contain the same information. Note that for a discrete r.v., the cdf is piecewise constant. Draw picture) 4. F X ( a ) x i x i a p X ( x i ) Example 9 Bernoulli

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• Fall '08
• Stites,M
• Probability, Probability theory, CDF

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