9Derived_Distributions

9Derived_Distributions - Chapter 9 Functions and Derived...

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Chapter 9 Functions and Derived Distributions We already know from our previous discussion that it is possible to form new discrete random variables by applying real-valued functions to existing random variables. In a similar manner, it is possible to generate a new random variable Y by taking a well-behaved function g ( · ) of a continuous random variable X . The graphical interpretation of this notion is analog to the discrete case and appears in Figure 9.1. Sample Space X Y = g ( X ) Figure 9.1: A function of a random variable is a random variable itself. In this ±gure, Y is obtained by applying function g ( · ) to the value of continuous random variable X . Let X be a continuous random variable, and let g : R ±→ R be an admis- sible function. Variable Y = g ( X ), de±ned pointwise, is itself random. The probability that Y falls in a speci±c set S depends on both the function g ( · ) 108
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9.1. MONOTONE FUNCTIONS 109 and the PDF of X , Pr( Y S ) = Pr( g ( X ) S ) = Pr ( X g - 1 ( S ) ) = ± g - 1 ( S ) f X ( ξ ) dξ. In particular, we can derive the CDF of Y using the formula F Y ( y ) = Pr( g ( X ) y )= ± { ξ X (Ω) | g ( ξ ) y } f X ( ξ ) dξ. (9.1) Example 73. Let X be a Rayleigh random variable with parameter σ 2 =1 , and defne Y = X 2 . We wish to fnd the distribution oF Y . ±rom (9.1) , we can compute the CD± oF Y . ±or y> 0 , we get F Y ( y ) = Pr( Y y ) = Pr ( X 2 y ) = Pr( - y X y ± y 0 ξe - ξ 2 2 = ± y 0 1 2 e - ζ 2 - e - y 2 . Above, we have used the Fact that X 0 in fnding the boundaries oF integra- tion, and we made the change oF variables ζ = ξ 2 in computing the integral. We recognize F Y ( · ) as the CD± oF an exponential random variable. This shows that the square oF a Rayleigh random variable possesses an exponential distri- bution. In general, the fact that X is a continuous random variable does not provide much information about the properties of Y = g ( X ). For instance, Y could be a continuous random variable, a discrete random variable or neither. To gain a better understanding of derived distributions, we begin our exposition of functions of continuous random variables by exploring speci±c cases. 9.1 Monotone Functions A monotonic Function is a function that preserves a given order. For instance, g ( · ) is monotone increasing if, for all x 1 and x 2 such that x 1 x 2 , we have g ( x 1 ) g ( x 2 ). Likewise, a function g ( · ) is termed monotone decreasing pro- vided that g ( x 1 ) g ( x 2 ) whenever x 1 x 2 . If the inequalities above are replaced by strict inequalities ( < and > ), then the corresponding functions
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110 CHAPTER 9. FUNCTIONS AND DERIVED DISTRIBUTIONS are said to be strictly monotonic . Monotonic functions of random variable are straightforward to handle because they allow a simple characterization of de- rived CDFs. For non-decreasing function g ( · ) of continuous random variable X , we have F Y ( y ) = Pr( Y y ) = Pr( g ( X ) y ) = Pr( g ( X ) ( -∞ ,y ]) = Pr( X g - 1 (( -∞ ])) = Pr ( X sup ± g - 1 (( -∞ ]) ²) = F X ( sup ± g - 1 (( -∞ ]) ²) .
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9Derived_Distributions - Chapter 9 Functions and Derived...

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