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# chap2f - Probability and Statistics with Reliability...

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Copyright © 2006 by K.S. Trivedi 1 Probability and Statistics with Reliability, Queuing and Computer Science Applications : second edition by K.S. Trivedi Publisher-John Wiley & Sons Chapter 2:Discrete Random Variables Dept. of Electrical & Computer engineering Duke University Email: [email protected] URL: www.ee.duke.edu/~kst

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Copyright © 2006 by K.S. Trivedi 2 Random Variables Sample space is often too large to deal with directly. Recall that in the sequence of Bernoulli trials, if we don’t need the detailed information about the actual pattern of 0’s and 1’s but only the number of 0’s and 1’s, we are able to reduce the sample space from size of ‘ 2 n to size of just ‘n+1’ . Such abstractions lead to the notion of a random variable.
Copyright © 2006 by K.S. Trivedi 3 Random Variables (cont’d) Discrete RV X: countable number of values. – Properties of a discrete RV: Probability mass function : p(x i ) of X Continuous RV X: uncountably infinite number of different values (an interval or a collection of intervals). (In Chapter 3) – Properties of a continuous RV: Probability density function : f(x) of X Note the distinction: pmf vs. pdf Notation : uppercase letters (X, Y) for RVs, lowercase, x,y, for the values.

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Copyright © 2006 by K.S. Trivedi 4 Discrete Random Variables A random variable (rv) X is a mapping (function) from the sample space S to the set of real numbers. i.e., a function that assigns a real number to each sample point Random variable is not a variable but a function If image of X (set of all values taken by X), is finite or countably infinite, X is a discrete rv.
Copyright © 2006 by K.S. Trivedi 5 Discrete Random Variables Inverse image A x of a real number x is the set of all sample points that are mapped by X into x : It is easy to see that the set of all inverse images are set that is mutually exclusive and collectively exhaustive:

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Copyright © 2006 by K.S. Trivedi 6 Discrete Random Variables (Example 2.1) Consider a random experiment defined by a sequence of three Bernoulli trials. The sample space S consists of eight triples of 0 s and 1 s Define a random variable X to be the total number of successes from three trials The values of the random variable X are {0,1,2,3} X(0,0,0) =0 X(0,0,1)=X(0,1,0)=X(1,0,0)=1 X(0,1,1)=X(1,0,1)=X(1,1,0) =2 X(1,1,1) = 3 The inverse images of random variable X are A 0 = {(0,0,0)} ; A 1 ={(0,0,1),(0,1,0),(1,0,0)}; A 2 ={(0,1,1),(1,1,0),(1,0,1)}; A 3 = {(1,1,1)}
Copyright © 2006 by K.S. Trivedi 7 Probability Mass Function (pmf) A x : set of all sample points such that, pmf is defined as = the probability that the value of the random variable X obtained on a performance of the experiment is equal to x

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Copyright © 2006 by K.S. Trivedi 8 Equivalence pmf: Probability mass function Discrete density function Sometimes mistakenly called the distribution function Empirical version is called a histogram