2. Rnd Var

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

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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: 7283, 86, 88, 90, 140144, 146150, 152157, 179186. 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 ( ) over the sample space of a random experiment, i.e., X : R X ( ) Randomness comes from the fact that outcomes are random ( X ( ) 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 = { 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 = R . Define the two random variables (a) X = (b) Y = +1 for - 1 otherwise 5. n packets arrive at a node in a communication network. Here 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 ( ) , { w : X ( ) A } inverse image of set A under X(w), i.e. {w:X(w)eA} R set A So, X A iff { w : X ( ) A } , thus P( { X A } ) = P( { w : X ( ) A } ) , or in short P { X A } = P { w : X ( ) 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 Definition...

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