Chapter 5 Random vectors, Joint distributions Lectures 18-23 In many real life problems, one often encounter multiple random objects. For example, if one is interested in the future price of two different stocks in a stock market. Since the price of one stock can affect the price of the second, it is not advisable to analysis them separately. To model such phenomenon, we need to introduce many random variables in a single platform (i.e., a probability space). First we will recall, some elementary facts about -dimensional Euclidean space. Let with the usual metric A subset of is said to be open if for each, there exists an such that where Any open set can be written as a countable union of open sets of the form, called open rectangles. Definition 5.1. The -field generated by all open sets in is called the Borel -field of subsets of and is denoted by. Theorem 5.0.16 Let Then Proof. We prove for, for, it is similar. Note that Hence from the definition of, we have Note that for,
For each such that we have Hence all open rectangles are in. Since any open set in can be rewritten as a countable union of open rectangles, all open sets are in. Therefore from the definition of, we get This completes the proof. (It is advised that student try to write down the proof for ) Definition 5.2. Let be a probability space. A map, is called a random vector if Now onwards we set (for simplicity) Theorem 5.0.17 is a random vector iff are random variables where denote the component of. Proof: Let For be a random vector. since Therefore is a random variable. Similarly, we can show that is a random variable. Suppose are random variables. For (5.0.1) Set By (5.0.1) (5.0.2)
For, we have Hence Thus. Similarly Hence Thus from (5.0.2), we have Therefore from Theorem 5.0.16, we have. Hence is a random vector. This completes the proof. Theorem 5.0.18 Let be a random vector. On define as follows Then is a probability measure on. Proof. Since, we have Let be pair wise disjoint elements from. Then are pair wise disjoint and are in. Hence This completes the proof. Definition 5.3. The probability measure is called the Law of the random vector and is denoted by. Definition 5.4. (joint distribution function)
Let be a random vector. Then the function given by is called the joint distribution function of. Theorem 5.0.19 Let be the joint distribution function of a random vector. Then satisfies the following. (i) (a) (b) (ii) is right continuous in each argument. (iii) is nondecreasing in each arguments. The proof of the above theorem is an easy exercise to the student. Given a random vector, the distribution function of denoted by is called the marginal distribution of. Similarly the marginal distribution function of is defined. Given the joint distribution function of, one can recover the corresponding marginal distributions as follows. Similarly Given the marginal distribution functions of and, in general it is impossible to construct the joint distribution function. Note that marginal distribution functions doesn't contain information about the dependence of over and vice versa. One can characterize the independence of and in terms of its joint and marginal distributions as in the following theorem. The proof is beyond the scope of this course. Theorem 5.0.20 Let be a random vector with distribution function. Then and are independent iff Definition 5.5. (joint pmf of discrete random vector) Let be a discrete random vector, i.e, are discrete random variables. Define by Then is called joint pmf of.
Definition 5.6. (joint pdf of continuous random vector) Let be a continuous random variable (i.e., are continuous random variables) with joint distribution function. If there exists a function such that then is called the joint pdf of. Theorem 5.0.21 Let be a continuous random vector with joint pdf. Then Proof. Note that L.H.S of the equality corresponds to the law of. Let denote the set of all finite union of rectangles in. Then is a field (exercise for the student). Set Then are probability measures on and on Hence, using extension theorem, we have i.e., Example 5.0.34 Let be two random variables with joint pdf given by If denote the marginal pdfs of and respectively, then
Therefore Here means is normally distributed with mean and variance. Similarly, Therefore Also note that and are dependent since, see exercise. Theorem 5.0.22 Let be independent random variables with joint pdf. Then the pdf of is given by where denote the convolution of and and is defined as Proof. Let denote the distribution function of. Set
Therefore This completes the proof. Example 5.0.35 Let be independent exponential random variables with parameters and respectively. Then is given similarly. Now for, clearly. For, Conditional Densities. The notion of conditional densities are intended to give a quantification of dependence of one random variable over the other if the random variables are not independent. Definition 5.7. Let be two discrete random variables with joint pmf. Then the conditional density of given denoted by is defined as Intuitively, means the pmf of given the information about. Here information about means knowledge about the occurrence (or non occurrence) of for each. One can rewrite in terms of the pmfs as follows.
Definition 5.8. Let are continuous random variables with joint pdf. The conditional distribution of given is defined as Definition 5.9. If are continuous random variable and if denote the conditional density of given. Then for, Example 5.0.36 Let be uniform random variable over and be uniform random variable over. i.e., Note that the pdf of given is, i.e. Also Hence