CDF of the of Joint Uniform Distribution for k random variables.

Question

If i have K independent Random Variable:

$X_1,X_2,x_3,\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot, X_k$ What would be the CDF of the sum of their Joint Distribution? $f_{X_1+X_2+X_3+...+X_k} (z)$ z<=1.

I cant figure out what would be their respective density functions and integration limits.

Answer 1

By independence:$$F_{X_1,\dots,X_k}(x_1,\dots,x_k)=P(X_1\leq x_1,\dots,X_k\leq x_k)=$$$$P(X_1\leq x_1)\times\cdots\times P(X_k\leq x_k)=F_{X_1}(x_1)\times\cdots\times F_{X_k}(x_k)$$

Answer 2

If they are independent, then

$$Pr(X_1 \le x_1, \ldots, X_k \le x_k ) = \prod_{i=1}^k Pr(X_i \le x_i)$$

That is the joint CDF is just the product of individual CDF.