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.
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
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 x_{1} and x_{2}
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 x_{1} and x_{2}