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STA213lecture6.notes

STA213lecture6.notes - Todays outline pp 55-68 Re-cap from...

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Today’s outline: pp 55-68 Re-cap from last class More examples of transformations Expected values Moments and Moment Generating Functions Example 2.3.10 and Thms 2.3.11 and 2.3.12 will be skipped (the only thing you need to know is that existence of MGFs implies uniqueness). Re-cap last class Variable transformations, what happens to the cdf and pdf? Why are they important? Next chapter will cover common distributions linked through important transformations. Theorem 2.1.5 Let X have pdf f X ( x ) and let Y = g ( X ), where g is a monotone function. Suppose that f X ( x ) is continuous and g - 1 ( y ) has a cts derivative. Pdf of Y is f y ( y ) = ( f X ( g - 1 ( y )) d dy g - 1 ( y ) y ∈ Y 0 otherwise Perhaps easier to understand, compare with integration by substitution: Want to integrate R b a f X ( x ) dx . Substitute y = g ( x ), so that x = g - 1 ( y ). Then f ( x ) becomes f X ( g - 1 ( y )). The differential dx becomes d dy g - 1 ( y ). The integral becomes R d c f X ( g - 1 ( y )) d dy g - 1 ( y ) dy Example 2.1.9: Normal-Chi square Example 2.1.6: Inverse Gamma Expectation definition 2.2.1 The expected value or mean (often denoted by μ ) of a random variable g ( X ), denoted by Eg ( X ), is Eg ( X ) = R -∞ g ( x ) f X ( x ) if X is continuous x ∈X g ( x ) P ( X = x ) if X is discrete Examples Exponential mean (2.2.2) Binomial mean (2.2.3) Cauchy mean (2.2.4) Symmetric distributions have mean in the middle!
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