[Math] Showing it is a joint probability density function

probabilityprobability distributions

I have two random variables $X,Y$ with a joint density function $f_{X,Y}(x,y)=x+y$ if $(x,y)\in[0,1]\times [0,1]$ and otherwise $f_{X,Y}(x,y)=0$

I want to analyze this case in different cases, first of all, I want to show it is a probability density function. Well I think the best way is to calculate it $f_{X,Y}(x,y) = f_{Y\mid X}(y|x)f_X(x) = f_{X\mid Y}(x\mid y)f_Y(y)$ but how can we evaluate the condtional and marginal distributions ?

Another idea is to show $\int_x \int_y f_{X,Y}(x,y) \; dy \; dx= 1.$

In the next step I would like to determine the cumulative distributions $F_X(x), F_Y(y)$

Best Answer

  1. Yes, you should check that $$\int_0^1\int_0^1(x+y)\,dydx=1\,.$$
  2. By definition, $F_X(t)=P(X<t)$, and this can be expressed by the integral: $$F_X(t)=\int_0^t\int_0^1 (x+y)\,dydx\,.$$
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