2.4.4 Bivariate Probability Distributions

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### 2.4.4 Bivariate Probability Distributions

We now consider two r.v.'s and , where and . We ask what is the probability that the first r.v. falls within and the second r.v. falls within , which defines the bivariate pdf :

Using this operational definition of , let us multiply and divide by the quantity , where we assume ,

It is readily shown that satisfies the properties for a legitimate pdf given in Eq. (24) and Eq. (25), and we can interpret as follows:

The quantity is known as the marginal probability distribution function. Now define the quantity ,

As with , it can be shown that is a legitimate pdf and can be interpreted as follows:

The quantity is called the conditional pdf. The constraint that simply means that the r.v.'s and are not mutually exclusive, meaning there is some probability that both and will occur together. Note that if and are independent r.v.'s, then and reduce to the univariate pdf's for and :

and therefore for independent pdf's we find that the bivariate pdf is simply the product of the two univariate pdf's:

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