Let A=[2;∞[^2 and f ∈(R×R)^R such as f(x,y)=((χ_A (x,y))/(E(x)^(E(y)) )) where χ_A is the caracteristic function of A Prove that f is a density of a probability P


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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

	$\int \int_{A} f (x, y) d x d y =$ $\sum_{n ⩾ 2} . \sum_{m ⩾ 2} \frac{1}{n^{m}} = \sum_{n ⩾ 2} \frac{1}{n^{2}} . \frac{1}{(1 - \frac{1}{n})}$ $= Σ \frac{1}{n (n - 1)} = \sum_{n ⩾ 2} (\frac{1}{n - 1} - \frac{1}{n}) = \lim_{M \to \infty} \underset{2}{\sum^{M}} (\frac{1}{n - 1} - \frac{1}{n})$ $= \underset{M \to \infty}{\lim 1} - \frac{1}{M} = 1$ $\int_{A} f d A = 1$ $f > 0$ $f d e n s i t y o f p r o b a b i l i t y$
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