Statistics/Distributions/Continuous
A continuous statistic is a random variable that does not have any points at which there is any distinct probability that the variable will be the corresponding number.
General Properties
editCumulative Distribution Function
editA continuous random variable, like a discrete random variable, has a cumulative distribution function. Like the one for a discrete random variable, it also increases towards 1. Depending on the random variable, it may reach one at a finite number, or it may not. The cdf is represented by a capital F.
Probability Distribution Function
editUnlike a discrete random variable, a continuous random variable has a probability density function instead of a probability mass function. The difference is that the former must integrate to 1, while the latter must have a total value of 1. The two are very similar, otherwise. The pdf is represented by a lowercase f.
Special Values
editLet R be the set of points of the distribution.
The expected value for a continuous variable X with probability density function f is defined as .
More generally, the expected value of any continuously transformed variable g(X) with probability density function f is defined as .
The mean of a continuous or discrete distribution is defined as .
![{\displaystyle E[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e455a34363c03fc5df8208d8b81fa29e3cdd524e)
The variance of a continuous or discrete distribution is defined as .
![{\displaystyle E[(X-E[X]^{2})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62e702258475ed39e0c1d2adcf70037b11a9955f)
Expectations can also be derived by producing the Moment Generating Function for the distribution in question. This is done by finding the expected value . Once the Moment Generating Function has been created, each derivative of the function gives a different piece of information about the distribution function.
![{\displaystyle E[\exp(tX)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/551b062fdc21eea6d3d8c6092cf3650bd083a3b2)
= mean
= variance
= skewness
= kurtosis
![{\displaystyle {\frac {dE[\exp(tX)]}{dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee4d4de7d968658bf7c423fb4b871f802850aaa9)
![{\displaystyle {\frac {d^{2}E[\exp(tX)]}{dt^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ab428b9fc2d6de74a61b5a8a96871bdf6cc4a59)
![{\displaystyle {\frac {d^{3}E[\exp(tX)]}{dt^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/116a5182d24a980cea50dc18e4d190bbe1ef8a4c)
![{\displaystyle {\frac {d^{4}E[\exp(tX)]}{dt^{4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa9ead6641087902016c6dbf87b0a5f09c24f3c)