Let-A-2-2-and-f-R-R-R-such-as-f-x-y-A-x-y-E-x-E-y-where-A-is-the-caracteristic-function-of-A-Prove-that-f-is-a-density-of-a-probability-P-

Question Number 126039 by snipers237 last updated on 16/Dec/20

L e t A = [2; \infty [^{2} a n d f \in {(R \times R)}^{R} s u c h a s

f (x, y) = \frac{χ_{A} (x, y)}{E {(x)}^{E (y)}} w h e r e χ_{A} i s t h e c a r a c t e r i s t i c f u n c t i o n o f A

P r o v e t h a t f i s a d e n s i t y o f a p r o b a b i l i t y P

Answered by mindispower last updated on 18/Dec/20

\int \int_{A} f (x, y) d x d y =

\sum_{n ⩾ 2} . \sum_{m ⩾ 2} \frac{1}{n^{m}} = \sum_{n ⩾ 2} \frac{1}{n^{2}} . \frac{1}{(1 - \frac{1}{n})}

= Σ \frac{1}{n (n - 1)} = \sum_{n ⩾ 2} (\frac{1}{n - 1} - \frac{1}{n}) = \lim_{M \to \infty} \underset{2}{\sum^{M}} (\frac{1}{n - 1} - \frac{1}{n})

= \underset{M \to \infty}{\lim 1} - \frac{1}{M} = 1

\int_{A} f d A = 1

f > 0

f d e n s i t y o f p r o b a b i l i t y