A probability distribution completely characterises the probabilistic behaviour of a random variable.
It specifies the probability associated with each possible value or interval of values that the variable can adopt.
For discrete random variables, the probability mass function gives the exact probability that the variable equals a specific number.
Each individual mass value must be non-negative, and the sum of all mass values across the domain must equal one.
For continuous random variables, the probability of any single exact point is zero because the continuum contains infinite points.
Instead, continuous variables use a probability density function, where probabilities correspond to areas under the density curve.
The total area under the probability density curve across all real numbers must integrate to exactly one.
The cumulative distribution function evaluates the accumulated probability from negative infinity up to a given threshold value.
For continuous variables, the derivative of the cumulative distribution function yields the probability density function.
Distributions are summarised by their mathematical moments: mean measures location, variance measures spread, and skewness measures shape.
Parametric distributions use a small set of mathematical parameters to model complex real-world data patterns compactly.
Selecting an inappropriate distribution model can lead to inaccurate forecasts and severely underestimated operational risks.
Non-parametric methods avoid strict distributional formulas by estimating probability density curves directly from empirical data.
Kernel density estimation is a popular non-parametric method that places smooth probability bumps over each recorded observation.
A frequent beginner mistake is interpreting the height of a continuous density curve as an exact probability percentage.