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# lect02 - Lecture Notes 2 Random Variables Denition...

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Lecture Notes 2 Random Variables Definition Probability Mass Function (PMF) Cumulative Distribution Function (CDF) Probability Density Function (PDF) Functions of a Random Variable Application: Generation of Random Variables EE 278: Random Variables 2 – 1 Random Variable A random variable (r.v.) is a real-valued function X ( ω ) over a sample space Ω , i.e., X : Ω R Ω ω X ( ω ) Notations: We use upper case letters for random variables: X, Y, Z, Φ , Θ , . . . We use lower case letters for values of random variables: X = x means that random variable X takes on the value x , i.e., X ( ω ) = x where ω is the outcome EE 278: Random Variables 2 – 2

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Examples of Random Variables 1. Let the random variable X be the number of heads in n coin flips. The sample space is Ω = { H, T } n , the possible outcomes of n coin flips; then X { 0 , 1 , 2 , . . . , n } 2. Let Ω = R , the real numbers. Define random variables X and Y as follows: a. X ( ω ) = ω b. Y ( ω ) = +1 ω 0 - 1 otherwise 3. Packet arrival times in the interval (0 , T ] . Here Ω is the set of all finite length strings ( t 1 , t 2 , . . . , t n ) (0 , T ] * , where t 1 t 2 · · · t n . Define the random variable X to be n , the length of the string; then X { 0 , 1 , 2 , 3 , . . . } 4. Let X be the service time at a router. If the bu ff er is empty the packet is served immediately, i.e., X = 0 . If it is not empty, the service time X > 0 is a positive real number EE 278: Random Variables 2 – 3 Specifying a Random Variable Specifying a random variable means being able to determine the probability that X A for any Borel set A R , in particular, for any interval ( a, b ] To do so, consider the inverse image of A under X , i.e., { ω : X ( ω ) A } R set A inverse image of A under X ( ω ) Since X A i ff ω { ω : X ( ω ) A } , P( { X A } ) = P( { ω : X ( ω ) A } ) = P { ω : X ( ω ) A } Shorthand: P( { set description } ) = P { set description } EE 278: Random Variables 2 – 4
Discrete Random Variables A random variable is said to be discrete if P { X X} = 1 for some countable set X R , i.e., X = { x 1 , x 2 , . . . } (finite or infinite) Examples 1, 2b, and 3 on page 2-3 are discrete random variables In general, X ( ω ) partitions Ω into the sets { ω : X ( ω ) = x i } , for i = 1 , 2 , . . . Ω . . . . . . x 1 x 2 x 3 x n R In order to specify X , it su ffi ces to know P { X = x i } for every i EE 278: Random Variables 2 – 5 A discrete random variable is thus completely specified by its probability mass function (pmf) p X ( x ) = P { X = x } for every x X Clearly p X ( x ) 0 and x X p X ( x ) = 1 Note that p X ( x ) can be simply viewed as a probability measure over a discrete sample space (even though the original sample space may be continuous as in examples 2b and 3) The probability of any (Borel) set A R is given by P { X A } = x A X p X ( x ) Notation: We use X p X ( x ) or simply X p ( x ) to mean that the discrete random variable X has pmf p X ( x ) or p ( x ) EE 278: Random Variables 2 – 6

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Famous Discrete Random Variables Bernoulli : X Bern( p ) for 0 p 1 has the pmf p X (1) = p and p X (0) = 1 - p Geometric : X Geom( p ) for 0 p 1 has the pmf p X ( k ) = p (1 - p ) k - 1 , k = 1 , 2 , 3 , . . .
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lect02 - Lecture Notes 2 Random Variables Denition...

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