A continuous distribution assigns probabilities over ranges, not single points. For a truly continuous variable, the probability of any exact value is zero.
Core Mechanics
A continuous random variable (X) is described by a probability density function (pdf) (f(x)) where:
[ P(a \le X \le b) = \int_a^b f(x),dx ]
and total probability satisfies:
[ \int_{-\infty}^{\infty} f(x),dx = 1 ]
The cumulative distribution function (cdf) is:
[ F(x) = P(X \le x) ]
Economic Use Cases
Continuous distributions are used for wages, returns, demand shocks, and error terms in regression models. They allow analysts to compute probabilities, quantiles, and expected values needed for risk and policy decisions.
Practical Context
Choosing a distributional assumption affects inference quality. For example, assuming normality when tails are heavy can understate downside risk in finance or forecast intervals in macro models.