n16 - CS 70-2 Discrete Mathematics and Probability Theory...

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Unformatted text preview: CS 70-2 Discrete Mathematics and Probability Theory Spring 2009 Alistair Sinclair, David Tse Lecture 16 Multiple Random Variables and Applications to Inference In many probability problems, we have to deal with multiple r.v.s defined on the same probability space. We have already seen examples of that when we saw, for example, that computing the expectation and variance of a binomial r.v. X , it is easier to write it as a sum X = n i = 1 X i where X i represents the result of the i th trial. In inference problems, where we observe certain quantities and use the information to infer about other hidden quantities, multiple r.v.s arise naturally in the modeling of the situation. We will see some examples of such problems after we go through some of the basics in the handling of multiple r.v.s. Joint Distributions Consider two random variables X and Y defined on the same probability space. By linearity of expectation, we know that E ( X + Y ) = E ( X )+ E ( Y ) . Since E ( X ) can be calculated if we know the distribution of X and E ( Y ) can be calculated if we know the distribution of Y , this means that E ( X + Y ) can be computed knowing only the two individual distributions. No information is needed about the relationship between X and Y . This is not true if we need to compute, say, E (( X + Y ) 2 ) , e.g. as when we computed the variance of a binomial r.v. This is because E (( X + Y ) 2 ) = E ( X 2 ) + 2E ( XY ) + E ( Y 2 ) , and E ( XY ) depends on the relationship between X and Y . How can we capture such a relationship? Recall that the distribution of a single random variable X is the collection of the probabilities of all events X = a , for all possible values of a that X can take on. When we have two random variables X and Y , we can think of ( X , Y ) as a two-dimensional random variable, in which case the events of interest are X = a Y = b for all possible values of ( a , b ) that ( X , Y ) can take on. Thus, a natural generalization of the notion of distribution to multiple random variables is the following. Definition 16.1 (joint distribution) : The joint distribution of two discrete random variables X and Y is the collection of values { ( a , b , Pr [ X = a Y = b ]) : ( a , b ) A B } , where A and B are the sets of all possible values taken by X and Y respectively. This notion obviously generalizes to three or more random variables. Since we will write Pr [ X = a Y = b ] quite often, we will abbreviate it to Pr [ X = a , Y = b ] . Just like the distribution of a single random variable, the joint distribution is normalized , i.e. a A , b B Pr [ X = a , Y = b ] = 1 . This follows from noticing that the events X = a Y = b , a A , b B , partition the sample space....
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n16 - CS 70-2 Discrete Mathematics and Probability Theory...

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