The smoothness of the kernel density estimate is evident compared to the discreteness of the histogram, as kernel density estimates converge faster to the true underlying density for continuous random variables.
22.
If you know the pdfs of two continuous random variables, what is the pdf of the sum of those two variables ?-- talk ) 03 : 31, 21 May 2010 ( UTC)
23.
To understand the problem we need to recognize that a distribution on a continuous random variable is described by a density " f " only with respect to some measure " ? ".
24.
The concept of the conditional distribution of a continuous random variable is not as intuitive as it might seem : Borel's paradox shows that conditional probability density functions need not be invariant under coordinate transformations.
25.
In general though, the PMF is used in the context of discrete random variables ( random variables that take values on a discrete set ), while PDF is used in the context of continuous random variables.
26.
Note that it is not possible to define a density with reference to an arbitrary measure ( e . g . one can't choose the counting measure as a reference for a continuous random variable ).
27.
As mentioned above, if " Y " is a continuous random variable, it is not possible to define \ scriptstyle \ operatorname { E } ( X \ mid Y ) by this method.
28.
The corresponding formula for a continuous random variable with probability density function with finite or infinite support \ mathbb X on the real line is defined by analogy, using the above form of the entropy as an expectation:
29.
Whereas the " pdf " exists only for continuous random variables, the " cdf " exists for all random variables ( including discrete random variables ) that take values in \ mathbb { R } \,.
30.
The probability integral transform states that if X is a continuous random variable with cumulative distribution function F _ X, then the random variable Y = F _ X ( X ) has a uniform distribution on [ 0, 1 ].
