# difference in CDFs and pdfs of joint distribution of two random variables

We know that the joint probability function of two independent random variables is just the product of their respective pdfs. On the same lines, .can we say that if we multiply the cumulative density functions(CDFs) of those two random variables, the resulting function will be the CDF of joint distribution? Like f1 and f2 are the two pdfs of the independent random variable. The joint pdf will simply be f1f2. If F1 and F2 are their CDFs.Can we say that F=F1F2 represents their joint CDF?

• Short answer: yes. Commented Jun 12, 2020 at 21:49
• Thanks Adrian.Can you please suggest me any text regarding this concept which I can see. Commented Jun 12, 2020 at 22:02
• any book or any other source Commented Jun 12, 2020 at 22:03

Yeah. Its just a simple exercise: Suppose $$X$$ is a random variable distributed according to pdf $$g$$ and cdf $$G$$. Similarly $$Y$$ is a random variable distributed according to pdf $$h$$ and pdf $$H$$. Let $$f$$ be the pdf of the joint distribution of $$(X,Y)$$.
$$X$$ and $$Y$$ are independent: i.e. $$f(x,y)=g(x)h(y)$$ for all $$(x,y)\in X\times Y$$.
Then we have the following: \begin{align} F(x,y) &= Pr(X\leq x, Y\leq y)\\ & = \int_{X\leq x}\int_{Y\leq y}f(s,t)dtds\\ & = \int_{X\leq x}\int_{Y\leq y}g(s)h(t)dtds\\ & = \int_{X\leq x}g(s)H(y)ds\\ & = G(x)H(y) \end{align}