Suppose $X = X_1$ with probability $p$ and $X = X_2$ with probability $1– p$ , where $p \in (0,1)$, $X_1$ and $X_2$ are random variables with CDF’s $F_1(x)$ and $F_2(x)$ respectively. Find the CDF of $X$.
I don't know where to go to find the CDF of $X$. I don't know how to use the fact that $X = X_1$ with probability p and $X = X_2$ with probability $1– p$.
Recall that the CDF $F(x)$ of $X$ is defined by $$ F(x) = \textrm{Pr}(X \leq x). $$ Apply the definition of $X$, and we see that $$ F(x) = \textrm{Pr}[X \leq x] = p \textrm{Pr}[X_1 \leq x] + (1-p)\textrm{Pr}[X_2 \leq x] = pF_1(x) + (1-p)F_2(x). $$