Mixed Random Variable

Compact study note.

Summary

Mixed random variable has both point masses and continuous component. Its distribution is not described completely by only PMF or only PDF.^[1]

Prerequisites

• Random Variable
• Discrete Random Variable
• Continuous Random Variable

Notation and Assumptions

A CDF $F_X$ is mixed when it has jumps at some points and continuous growth over intervals. Jump size at point $c$ equals $P(X=c)$ .

Essential Result

Compute probabilities by adding point masses and integrating the density over the continuous part.

Small Example

Let $P(X=0)=0.3$ and, conditional on not being zero, $X$ be uniform on $(0,1)$ with weight $0.7$ . Then $P(X\le 0.5)=0.3+0.7(0.5)=0.65$ .

Common Mistakes

• Representing the continuous component with an impossible atom at infinity.
• Dropping the point mass when computing expectations.

Connections

• Cumulative Distribution Function
• Expected Value
• Probability Density Function

References