IEN310 Chapter 5 - Chapter 5 Random Vectors 5 Introduction...

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Chapter 5: Random Vectors 5. Introduction The definitions and theorems in Chapter 4 concerning probability models of two random variables X and Y can be generalized to experiments that yield an arbitrary number of ran- dom variables X 1 , . . . , X n . Vector notation provides a convenient way of treating a collection of n random variables as a single entity. Vector and matrix notation will thus be used in this chapter to provide a concise representation of probability models and their properties. 5.1 Probability Models of N Random Variables Definition 5.1 Multivariate Joint CDF : The joint cumulative distribution function of ran- dom variables X 1 , . . . , X n is F X 1 ,...,X n ( x 1 , . . . , x n ) = P [ X 1 x 1 , . . . , X n x n ] . Although the multivariate joint CDF provides a complete probability model regardless of whether any or all of the X i are discrete, continuous, or mixed, it is usually not conveneient to use in analyzing probability models. Instead, the joint PMF or the joint PDF is used. Definition 5.2 Multivariate Joint PMF : The joint PMF of the discrete random variables X 1 , . . . , X n is P X 1 ,...,X n ( x 1 , . . . , x n ) = P [ X 1 = x 1 , . . . , X n = x n ] . Definition 5.3 Multivariate Joint PDF : The joint PDF of the continuous random variables X 1 , . . . , X n is the function f X 1 ,...,X n ( x 1 , . . . , x n ) = n F X 1 ,...,X n ( x 1 , . . . , x n ) ∂x 1 · · · ∂x n . Theorem 5.1 If X 1 , . . . , X n are discrete random variables with joint PMF P X 1 ,...,X n ( x 1 , . . . , x n ), P X 1 ,...,X n ( x 1 , . . . , x n ) 0 , X x 1 S X 1 · · · X x n S X n P X 1 ,...,X n ( x 1 , . . . , x n ) = 1 . 77
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Theorem 5.2 If X 1 , . . . , X n are continuous random variables with joint PDF f X 1 ,...,X n ( x 1 , . . . , x n ), then f X 1 ,...,X n ( x 1 , . . . , x n ) 0 , F X 1 ,...,X n ( x 1 , . . . , x n ) = Z x 1 -∞ · · · Z x n -∞ f X 1 ,...,X n ( u 1 , . . . , u n ) du 1 · · · du n , Z -∞ · · · Z -∞ f X 1 ,...,X n ( x 1 , . . . , x n ) dx 1 · · · dx n = 1 . In many situations, an event A is described in terms of a property of X 1 , . . . , X n (for example, A is the event such that max i X i 100). To find the probability of the event A , we sum the joint PMF or integrate the joint PDF over all x 1 , . . . , x n that belong to A , as stated in the following theorem. Theorem 5.3 The probability of an event A expressed in terms of the random variables X 1 , . . . , X n is, in the discrete case , P [ A ] = X ( x 1 ,...,x n ) A P X 1 ,...,X n ( x 1 , . . . , x n ) , (where the single summation actually represents a multiple sum over the n random variables) and, in the continuous case , P [ A ] = Z · · · Z A f X 1 ,...,X n ( x 1 , . . . , x n ) dx 1 · · · dx n . Example 1 (page 213) Consider a set of n independent trials in which there are r possible outcomes s 1 , . . . , s r for each trial. In each trial, P [ s i ] = p i . Let N i equal the number of times that outcome s i occurs over n trials. What is the joint PMF of N 1 , . . . , N r ? It turns out that the solution to this problem appeared in Theorem 1.19, which used the multinomial coefficient . We now see that the solution to this problem actually represents a type of joint PMF known as the multinomial distribution : P [ N 1 , . . . , N r ] = n n 1 , . . . , n r !
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