What does each function tell us?
The cumulative distribution function (CDF) answers “What is the probability that X is at most x?” The probability density function (PDF) describes how densely probability concentrates near x for an absolutely continuous variable. Its height is not a point probability.
The interval identity applies to every distribution. For continuous variables, the choice to include an endpoint makes no difference. For discrete variables it can matter.
A compact comparison
| Question | Function | Interpretation |
|---|---|---|
| Where is probability concentrated? | Probability density function | Density at x, measured per unit of x |
| How much probability has accumulated? | Cumulative distribution function | A number between 0 and 1 |
| What is the chance of exactly k defaults? | Probability mass function (PMF) | P(X = k) for a discrete variable |
| Which threshold contains p of the outcomes? | Inverse cumulative distribution function | A quantile or percentile |
Worked example: a standard normal variable
Let Z have mean 0 and standard deviation 1. The density at zero is about 0.39894. The cumulative probability at zero is 0.50.
| Quantity | Value | Meaning |
|---|---|---|
| f(0) | 0.39894 | Height of density at zero |
| F(0) | 0.50000 | 50% probability at or below zero |
| F(1) − F(−1) | 0.68269 | About 68.27% between −1 and +1 |
| P(Z = 0) | 0 | A single point has no area |
On the chart, switch between density and cumulative probability. The density rises and falls. The cumulative curve never falls.
Discrete cumulative probabilities have jumps
For two independent loans with 10% default probability each, the default-count masses are 0.81, 0.18 and 0.01. Thus F(0) = 0.81, F(1) = 0.99 and F(2) = 1. Between these integer values the cumulative function remains flat.
Every cumulative distribution function is non-decreasing and right-continuous. Its limiting values are 0 at the far left and 1 at the far right. A jump at x equals P(X = x).
Quantiles and tail questions
The general p-quantile is the smallest x such that F(x) ≥ p. For a standard normal variable, the 95th percentile is approximately 1.64485. About 95% of outcomes lie at or below this value.
For a loss variable, a high percentile represents a large loss. For a return variable, a low percentile represents a poor return. State the sign convention before reporting a risk number.
Excel: TRUE and FALSE
| Task | Formula |
|---|---|
| Normal cumulative probability at 1 | =NORM.S.DIST(1,TRUE) |
| Normal density at 1 | =NORM.S.DIST(1,FALSE) |
| Probability between −1 and 1 | =NORM.S.DIST(1,TRUE)-NORM.S.DIST(-1,TRUE) |
| 95th percentile | =NORM.S.INV(0.95) |
TRUE selects cumulative probability in these functions. FALSE selects density for a continuous distribution, but probability mass in BINOM.DIST.
Check your understanding
Is F(2) − F(1) the probability at 2?
No. It is P(1 < X ≤ 2). A point probability is a jump in the cumulative function, rather than the difference across an interval of positive width.
Interactive distribution laboratory
Change the parameters and threshold, then switch between cumulative probability and density or mass. Values are numerical approximations for learning.
Horizontal axis: the variable’s value. Vertical axis: density, probability mass, or cumulative probability as labelled. Shading highlights values at or below the threshold. The plot shows a finite viewing window.
Sources and further reading
- National Institute of Standards and Technology: Gallery of Distributions
- Distribution definition and properties
- Microsoft: Statistical functions reference
All numerical finance and project examples are hypothetical teaching illustrations. Probability assumptions require validation before practical use.