# Continuous Distribution

Continuous probability distribution combines the possible values of a continuous random variable with corresponding probabilities. While theoretical and discrete probabilities are used in discrete distributions, probability density function f(x) is used for computing probabilities in continuous distributions. The pdf of a continuous random variable X f(x) ≥ 0 for all x. The probabilities for the distribution are calculated using definite integral as

P( a ≤ X ≤ b) = $\int_{a}^{b}f(x)dx$ and the probability for the definite outcome is given by $\int_{-\infty }^{\infty }f(x)dx$ = 1.

As a continuous variable can assume values only as intervals, the probability of the variable assuming a specific value in the interval is zero.

P( X = c) = $\int_{c}^{c}xf(x)dx$ = 0.

The mean and variance of a continuous probability distribution is given by,

Mean μ = Expectation E(x) = $\int_{-\infty }^{\infty }xf(x)dx$ and Var (X) = [E(x)]^{2} - μ^{2}.