lect02

# lect02 - Lecture Notes 2 Random Variables • Definition...

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Unformatted text preview: 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 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 ∈ { , 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 ≥- 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 ∈ { , 1 , 2 , 3 , . . . } 4. Let X be the service time at a router. If the buffer is empty the packet is served immediately, i.e., X = 0 . If it is not empty, the service time X > 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 iff 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 suffices 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 ) ≥ 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...
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## This note was uploaded on 04/07/2010 for the course EE 278 taught by Professor Balajiprabhakar during the Spring '09 term at Stanford.

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lect02 - Lecture Notes 2 Random Variables • Definition...

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