The distribution of a continuous random variable can be characterized through its probability density function (pdf). The probability that a continuous random variable takes a value in a given interval is equal to the integral of its probability density function over that interval, which in turn is equal to the area of the region in the xy-plane bounded by the x-axis, the pdf and the vertical lines corresponding to the boundaries of the interval.
For example, in the picture below the blue line is the pdf of a normal random variable and the area of the red region is equal to the probability that the random variable takes a value comprised between -2 and 2.
The following is a formal definition.
Definition The probability density function of a continuous random variable is a function such thatfor any interval .
The set of values for which is called the support of .
Suppose a random variable has probability density function
To compute the probability that takes a value in the interval ,you need to integrate the probability density function over that interval:
Related concepts are those of joint probability density function, which characterizes the distribution of a continuous random vector, marginal probability density function, which characterizes the distribution of a subset of entries of a random vector, and conditional probability density function, which is a pdf obtained by conditioning on the realization of another random variable.
Probability density functions are discussed in more detail in the lecture entitled Random variables.
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