Let us assume that $Y$ is a discrete r.v. while $Z$ is continuous. $ X= \begin{cases} Y & \text{w.p.} && p \\ Z & \text{w.p.} && (1-p) \\ \end{cases} $ Such a r.v. $X$ has a mixed distribution. - It is *not discrete*, since the continuous part cannot be mapped with an index. - It is *not continuous*, since that requires single points to have $0$ probability. Consequently these mixed distributions do not have a valid [[Probability Mass Function]] or [[Probability Density Function]]. However they always have a [[Cumulative Density Function]]. $ \begin{align} F_X(x)&=p*\mathbf P(Y \le x)+(1-p)*\mathbf P(Z \le x) \\[2pt] F_X(x)&=p*F_Y(x)+(1-p)*F_Z(x) \end{align} $