# 4. Continuous probability distributions

If the sample space of a random variable x coincides with the set of the real numbers, a real continuous function p(x) such that is said a probability density function.

The probability of an event [a,b] is given by Generalizing the definitions given in the previous section for discrete distributions

• the mean value of x is given by • the mean value of the squares is • the variance is Example.

If p(x) is defined as it is a probability density. In fact, in the range in which it is greater than zero, it is graphically represented by a semicircle of radius . The area of this semicircle, and then the integral from -∞ to +∞, is 1.

## Uniform continuous distribution

A function p(x) such that is said a uniform continuous distribution

The mean value is The mean of the squares is The variance is and the standard deviation ## Exponential continuous distribution

Given a real positive value λ(lambda), the function p(x) such that is said a exponential continuous distribution.

This function is always > 0 and • The mean value is Integrating by parts  • The mean value of the squares is Integrating by parts • The variance is and the standard deviation The following JavaScript application allows you to calculate and to graph an exponential distribution. To view the tables, your browser must allow popups.

The following JS application allows to calculate the probability that in an exponential distribution with prefixed λ an event takes values between x1 and x2

## Gaussian continuous distribution

Given two real positive values A and a, a function p(x) such that is a probability density function if Since p(x) is an even function, this equality is equivalent to the following We can demonstrate that so Then p(x) depends only on the parameter a Since p(x) is an even function, its mean value is 0 and its variance is We can demonstrate that then Now we can write p(x) directly in terms of its variance The equality (4.22) is said a gaussian distribution. The graph of this curve is the well known bell curve, symmetrical about the y-axis, with a maximum at x=0 and inflexion points at ±σ.

If we translate the curve of a quantity μ, its equation is expressed by The following JavaScript application allows you to calculate and to graph a gaussian distribution. To view the tables, your browser must allow popups.

The following JS application allows you to calculate the probability that in an exponential distribution, with mean 0 and prefixed σ, an event takes values between x1 and x2