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# 3.1-3.4 - Chapter 3 Discrete Random Variables In this...

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Chapter 3. Discrete Random Variables In this section we discuss the following: 1 Random Variable 2 Probability Distributions 3 Mean and variance of a random variable 4 Calculations with random variables Definition . A random variable is a function that assigns numerical values to physical outcomes of a random experiment. A random variable is typically represented by X or x . That is, X : S R Thus, if ) ( then , e x S e is a real value. Random variable is discrete if its range is a finite or countable set. Random variable is continuous if its range is an interval on the real line. 1

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Examples 1. some examples are: x = outcome of a die rolled (discrete) x = Number of students passing a test (discrete) x = amount of time a student studies (continuous) x = lifetime of a bulb (continuous) Discrete random variable Let X be a discrete variable taking values in {0, 1, 2, …} = S. Definition : The distribution of discrete rv X is given by probability mass function p(x)=P(X=x)=P(all s S: X(s)=x) satisfying (i) p(x) ≥ 0 for all x S (ii) S x p(x) =1 = [ p(0) + p(1) + ….= 1] This denotes the list of possible values of discrete random variables X and their associated probabilities. Example 2. Three tosses of a fair coin. Define X = the number of heads in three tosses of a fair coin. Then, S = { HHH, HHT, HTH, THH, TTH, THT, HTT, TTT} And X= 3 2 2 2 1 1 1 0 2
The probability distribution is given by X f (x) 0 1/8 1 3/8 2 3/8 3 1/8 Example 3.

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