Advanced Concept

Mixed Random Variables

Not purely discrete, not purely continuous. A mix of smooth curves and sudden jumps.

1 What is a Mixed Random Variable?

A Mixed Random Variable is a combination of discrete and continuous parts. Its Cumulative Distribution Function (CDF) has both:

  • Continuous segments: Where probability accumulates smoothly.
  • Jumps (Discontinuities): Where there is a non-zero probability at a specific point (like a discrete variable).
$$ F_Y(y) = C(y) + D(y) $$

Sum of Continuous part C(y) and Discrete part D(y)

2 Visualizing the CDF

Consider a variable \( Y \) that behaves like \( X^2 \) for \( 0 \le X < 0.5 \), but then jumps to a fixed value. The graph below shows a CDF with a smooth curve and a sudden vertical jump.

Interactive Mixed CDF

Adjust the "Jump Size" to see how the discrete part affects the total CDF. The remaining probability is distributed continuously.

3 Decomposition

Continuous Part

Represents the probability density over intervals.

$$ \int_{-\infty}^{\infty} c(y) \, dy < 1 $$

Discrete Part

Represents the probability mass at specific points (jumps).

$$ \sum P(Y=y_k) > 0 $$
The sum of the integral of the continuous part and the sum of the discrete probabilities must equal 1.