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We have a bivariate random variable with the joint PDF $$f(x_1,x_2)=0.5I_{[0,4]}x_1 I_{[0,0.25x_1]}x_2$$ then find the joint CDF ?

It is a homework and I know how can I find the joint CDF , but I do not understand how can this function be a valid PDF ? , since $x_1 \in [0,4] $ and $x_2 \in [0,0.25x_1]$ if we integrate over the entire domain we must get $1$ but I get $2$

$$\int_0^4 \int_0^1 0.5 \, dx_2 \, dx_1 = 2 $$ any help please !

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The correct integral is

$$ \int_0^4 \int_0^{x_1 / 4} \frac{1}{2} \, \mathrm d x_2 \,\mathrm dx_1 $$

which does, in fact, integrate up to $1$.

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