Question

(1) difficult. If the kernel without the indicator function is r-d where d > 1, how is X distributed? (5) (difficult Prove B(0, 3) = For not required. using the method from class (ie the textbook) is Problem 3 We will now practice using order statistics concepts. (a) JeasyIf X...., x, f(x) where its CDF is denoted Fx), express the CDF of the maximum X, and express the CDF of the minimum x.

Answer 1

Lets assume f(x) to be an exponential distribution.

Given, this is order statistics -

=> X1 = min(X1, X2 , ...Xn)

and Xn = max(X1, X2 , ...Xn)

and, X1 < X2 < X3 .... <Xn

Given, we have to find the cdf, so x must be continuous.

The cdf of X1 -

P (X1 > x) = ^{i = 1}$\prod$_{i = N} P (Xi > x)

= ^{i = 1}$\prod$_{i = N} [1 - F(x)]

= [1 - F(x)]ⁿ

Fx1(x) = 1 - [1 - F(x)]ⁿ

fx1(x) = n[1 - F(x)]^n-1f(x) { Since, f(x) is the differential of F(x) }

Since, f(x) = $\lambda$e^-$\lambda$x, x>0 (exponential distribution)

=> fx1(x) =  n$\lambda$e^-$\lambda$x [1 - (1 - e^-$\lambda$x)]^n-1

= n$\lambda$(e^-$\lambda$x)ⁿ

Whereas the cdf of Xn -

Fxn(x) = P (Xn <= x )

= ^{i = 1}$\prod$_{i = N} P (Xi <= x)

= F(x)ⁿ

fxn(x) = n[F(x)]^n-1 f(x)

= n [1 - e^-$\lambda$x]^n-1$\lambda$e^-$\lambda$x