2. Rnd Var

2. Rnd Var - Lecture Notes 2 Random Variables •...

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Unformatted text preview: Lecture Notes 2 Random Variables • Definition • Discrete Random Variables: Probability mass function (pmf) • Continuous Random Variables: Probability density function (pdf) • Mean and Variance • Cumulative Distribution Function (cdf) • Functions of Random Variables Corresponding pages from B&T textbook: 72–83, 86, 88, 90, 140–144, 146–150, 152–157, 179–186. EE 178: Random Variables Page 2 – 1 Random Variable • A random variable is a variable that takes on values randomly Sounds nice, but not terribly precise or useful • Mathematically, a random variable (r.v.) X is a real-valued function X ( ω ) over the sample space Ω of a random experiment, i.e., X : Ω → R Ω ω X ( ω ) • Randomness comes from the fact that outcomes are random ( X ( ω ) is a deterministic function) • Notations: ◦ Always use upper case letters for random variables ( X , Y , . . . ) ◦ Always use lower case letters for values of random variables: X = x means that the random variable X takes on the value x EE 178: Random Variables Page 2 – 2 • Examples: 1. Flip a coin n times. Here Ω = { H, T } n . Define the random variable X ∈ { , 1 , 2 , . . . , n } to be the number of heads 2. Roll a 4-sided die twice. (a) Define the random variable X as the maximum of the two rolls 2nd roll 1st roll 1 2 3 4 1 2 3 4 Real Line 1 2 3 4 EE 178: Random Variables Page 2 – 3 (b) Define the random variable Y to be the sum of the outcomes of the two rolls (c) Define the random variable Z to be if the sum of the two rolls is odd and 1 if it is even 3. Flip coin until first heads shows up. Define the random variable X ∈ { , 1 , 2 . . . . } to be the number of flips until the first heads 4. Let Ω = R . Define the two random variables (a) X = ω (b) Y = +1 for ω ≥- 1 otherwise 5. n packets arrive at a node in a communication network. Here Ω is the set of all arrival time sequences ( t 1 , t 2 , . . . , t n ) ∈ (0 , ∞ ) n . (a) Define the random variable N to be the number of packets arriving in the interval (0 , 1] (b) Define the random variable T to be the first interarrival time EE 178: Random Variables Page 2 – 4 Specifying a Random Variable • Specifying a random variable means being able to determine the probability that X ∈ A for any event A ⊂ R , e.g., any interval • To do so, we consider the inverse image of the set A under X ( ω ) , { w : X ( ω ) ∈ A } inverse image of set A under X(w), i.e. {w:X(w)eA} R set A • So, X ∈ A iff ω ∈ { w : X ( ω ) ∈ A } , thus P( { X ∈ A } ) = P( { w : X ( ω ) ∈ A } ) , or in short P { X ∈ A } = P { w : X ( ω ) ∈ A } EE 178: Random Variables Page 2 – 5 • Example: Roll fair 4-sided die twice independently: Define the r.v. X to be the maximum of the two rolls. What is the P { . 5 < X ≤ 2 } ?...
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2. Rnd Var - Lecture Notes 2 Random Variables •...

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