By James E. Gentle

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8 distribution of the minimum order statistic in a twoparameter exponential distribution Consider a shifted version of the exponential family of distributions, called the two-parameter exponential with parameter (α, θ). Suppose the random variables X1 , . . d. with Lebesgue PDF pα,θ (x) = θ−1 e−(x−α)/θ I]α,∞[ (x). We want to find the distribution of the minimum of {X1 , . . , Xn }. Let us denote that minimum by Y . (This is an order statistic, for which we use a general notation, Y = X(1) . ) We have Pr(Y ≤ t) = 1 − Pr(Y > t) = 1 − Pr(Xi > t for i = 1, .

Therefore the CDF FY of Y is FY (x) = Pr(Y −1 [] − ∞, x]]) √ √ = Pr(X −1 [[− x, x]]) √ √ = (FX ( x) − FX (− x))I¯IR+ (x). Differentiating, we have the Lebesgue PDF of Y : √ √ 1 fY (x) = √ (fX ( x) + fX (− x))I¯IR+ (x). 85) Method of Change of Variables If X has density pX (x|α, θ) and Y = h(X), where h is a full-rank transformation (that is, there is a function h−1 such that X = h−1 (Y )), then the density of Y is A Companion for Mathematical Statistics c 2010 James E. 86) where Jh−1 (y) is the Jacobian of the inverse transformation, and | · | is the determinant.

9 distribution of the square of a random variable 2 Given X with CDF FX and Lebesgue PDF f√ X . Let √ Y = X . For x < 0, −1 −1 −1 Y []−∞, x]] = ∅, and Y []−∞, x]] = X [[− x, x]], otherwise. Therefore the CDF FY of Y is FY (x) = Pr(Y −1 [] − ∞, x]]) √ √ = Pr(X −1 [[− x, x]]) √ √ = (FX ( x) − FX (− x))I¯IR+ (x). Differentiating, we have the Lebesgue PDF of Y : √ √ 1 fY (x) = √ (fX ( x) + fX (− x))I¯IR+ (x). 85) Method of Change of Variables If X has density pX (x|α, θ) and Y = h(X), where h is a full-rank transformation (that is, there is a function h−1 such that X = h−1 (Y )), then the density of Y is A Companion for Mathematical Statistics c 2010 James E.

### A Companion for Mathematical Statistics by James E. Gentle

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