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STA213lecture5.notes

STA213lecture5.notes - Todays outline pp 47-55 Examples of...

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Today’s outline: pp 47-55 Examples of variable transformation One-to-one transformations Theorems Example 1: Exponential dist’n [similar to HW 1.53, 1.55] We are given a random variable X with pdf f X ( x ) = λ exp( - λx ) , x > 0 . We say that X Exp ( λ ) is exponentially distributed with parameter (also called rate) λ . What is the cdf of X ? What is the pdf of Y = 2 X ? How about Z = λX ? P ( Y < y ) = P (2 X < y ) = P ( X < y/ 2) = Z y/ 2 0 f X ( x ) dx = 1 - exp( - λy/ 2) , P ( Z < z ) = 1 - exp( - z ) . Differentiating, the pdf of Z becomes g ( z ) = exp( - z ) So Y and Z are also exponentially distributed, with parameters λ/ 2 and 1, respectively. Example 2: Exponential distribution As before, X with pdf f X ( x ) = λ exp( - λx ) , x > 0 . Take Y = X 2 . What is the cdf of Y ? P ( Y < y ) = P ( X 2 < y ) = P ( X < y ) = 1 - exp( - λ p ( y )) . The pdf becomes f Y ( y ) = exp( - λ y ) 2 y . Example 3: Double exponential (HW 1.54(b)) Now consider r.v. X with pdf f X ( x ) = 1 2 λ exp( - λ | x | ) , -∞ < x < . What does this look like?

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