Marginal Distribution

If random vector $\bar{X}$ has a joint c.d.f $F_{\bar{X}}(\bar{x})$, denoted $\bar{X} \sim F_{\bar{X}}(\bar{x})$ then each of $X_i's$ has a p.d.f (p.m.f) and c.d.f called \textbf{marginal p.d.f}( denoted $f_{X_i}$ ) and \textbf{marginal c.d.f}( denoted $F_{X_i}$ ). $F_{X_i}(x) = \begin{cases} \sum_{\bar{x} | \bar{x_i} \leq x} f_{\bar{X}}(\bar{x}) & \text{if }\bar{X}\text{ is discrete} \int_{-\infty}^{\infty}... \int_{-\infty}^{x_i}...\int_{-\infty}^{\infty} f_{\bar{X}}(\bar{x}) \mathrm{d}x_n ...\mathrm{d}x_i...\mathrm{d}x_1 & \text{if }\bar{X}\text{ is continuous} \end{cases}$ $f_{X_i}(x) = \begin{cases} \sum_{\bar{x} | \bar{x_i} = x} f_{\bar{X}}(\bar{x}) & \text{if }\bar{X}\text{ is discrete} F_{X_i}'(x) & \text{if }\bar{X}\text{ is continuous} \end{cases}$

If you recall the definition of independent random variable, you'll see that if $X_1, X_2, ..., X_n$ are independent random variable, the joint c.d.f. and p.d.f can be calculate as term-by-term product, i.e, $F_{\bar{X}}( \bar{x} ) = F_{X_1}(x_1)F_{X_2}(x_2)...F_{X_n}(x_n)$ $f_{\bar{X}}( \bar{x} ) = f_{X_1}(x_1)f_{X_2}(x_2)...f_{X_n}(x_n)$