Everything about Continuous Probability Distribution totally explained
In
probability theory, a
probability distribution is called
continuous if its
cumulative distribution function is
continuous. That is equivalent to saying that for
random variables
X with the distribution in question, Pr[
X=
a] = 0 for all
real numbers
a, for example: the probability that
X attains the value
a is zero, for any number
a. If the distribution of
X is continuous then
X is called a
continuous random variable.
While for a
discrete probability distribution one could say that an
event with
probability zero is impossible, this can't be said in the case of a continuous random variable, because then no value would be possible. This
paradox is resolved by realizing that the probability that
X attains some value within an
uncountable set (for example an
interval) can't be found by adding the probabilities for individual values.
Under an alternative and stronger definition, the term "continuous probability distribution" is reserved for distributions that have
probability density functions. These are most precisely called
absolutely continuous random variables (see
Radon–Nikodym theorem). For a random variable X, being absolutely continuous is equivalent to saying that the probability that X attains a value in any given
subset S of its range with
Lebesgue measure zero is equal to zero. This doesn't follow from the condition Pr[
X=
a] = 0 for all real numbers
a, since there are uncountable sets with Lebesgue-measure zero (for example the
Cantor set).
A random variable with the
Cantor distribution is continuous according to the first convention, but according to the second, it isn't (absolutely) continuous. Also, it isn't discrete nor a weighted average of discrete and absolutely continuous random variables.
In practical applications, random variables are often either discrete or absolutely continuous, although mixtures of the two also arise naturally.
The
normal distribution,
continuous uniform distribution,
Beta distribution, and
Gamma distribution are well known absolutely continuous distributions. The normal distribution, also called the Gaussian or the bell curve, is ubiquitous in nature and statistics due to the
central limit theorem: every variable that can be modelled as a sum of many small independent variables is approximately normal.
Further Information
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