Mathematical Modelling Question

Hummingbird25

Hi

Given [tex]X_1 \ldots X_n[/tex] be stochastic independent variables with the distribution functions [tex]F_X_{1}, \ldots ,F_X_{n} [/tex]. [tex]U = min(X_1 \ldots X_n)[/tex] and [tex]V = min(X_1 \ldots X_n)[/tex].

[tex]F_{U}[/tex] and [tex]F_{V}[/tex] for U and V, and let [tex]F_{U,V}[/tex] be simultaneously distribution functions for the stochastic vectors (U,V).

Then show that

[tex]F_{V} (s) = \Pi \limit_{i=1} ^{n} F_{X_i} (s)[/tex] where [tex]\forall s \in \mathbb{R}[/tex]

I can see that if I expand the sum I get

[tex]F_X_{1}(s) + F_X_{2}(s) + F_X_{3}(s) + \ldots + F_X_{i}(s)[/tex] where [tex]1 \leq i \leq n [/tex]

Doesn't that mean that

[tex]F_X_{1}(s) + F_X_{2}(s) + F_X_{3}(s) + \ldots + F_X_{i}(s) = (F_X_{1}(s) \ \mathrm{U} \ F_X_{2}(s) \ \mathrm{U} F_X_{3}(s) \ \mathrm{U} \ \ldots \ \mathrm{U} \ F_X_{n}(s))[/tex] ??

Since [tex]\sum_{i=1} ^{n} P(A_i) = P(A_1) + P(A_2) + P(A_3) + \ldots + P(A_n) [/tex]

Sincerely
Hummingbird

HallsofIvy

May we assume that
[tex]V = min(X_1 \ldots X_n)[/tex]
was actually supposed to be
[tex]V = max(X_1 \ldots X_n)[/tex]

Hummingbird25

My assignment uses U and V to distingues between min and max, but I guess it doesn't make that a bit a difference in the final calculation.

Sincerely Humingbird