Mixed Distributions
A mixed-type distribution has both a continuous part and one or more point masses. Its CDF is the sum:
\[F(x) = W_c \cdot G(x) + \sum_{x_i \leq x} w_i,\]where $G$ is the continuous CDF (weight $W_c$) and the $w_i$ are the point masses at $x_i$. The defining visual feature: the CDF picks up a jump of size exactly $w_i$ at every $x_i$, while remaining smooth in between.
Below, pick a continuous family and add point masses with weights $w_i$. The continuous weight $W_c = 1 - \sum w_i$ is allocated automatically — what’s left over after the discrete part. Sweep current $x$ and watch the CDF jump as you cross each $x_i$.
Interactive
| $x_i$ | $w_i$ |
|---|
What it shows
Jumps are the signature of mixed type. A purely continuous CDF is smooth; a purely discrete CDF is all jumps; a mixed CDF has both. The jump at $x_i$ has height exactly $w_i$ — the same value you typed into the table. Slide current $x$ across an $x_i$ and watch the CDF lift by $w_i$ in one step, while the corresponding lollipop in the upper panel turns blue.
Continuous and discrete share the budget. Adding a point mass $w_i$ removes $w_i$ from the continuous part: the smooth curve scales down by $W_c = 1 - \sum w_j$, and the lost area under the density reappears as the jumps. Total probability stays at 1 by construction.
Mixed PDF is not a function in the ordinary sense. Strictly, $f_{\text{mix}}(x) = W_c\, g(x) + \sum_i w_i\, \delta(x - x_i)$, with Dirac deltas at the mass points. The lollipops are just a visual stand-in for those deltas — heights of lollipops carry probability mass (unitless), heights of the smooth curve carry probability density (per unit $x$). They share an axis here for layout convenience, not because they share units.