Why is the CDF for the normal distribution can not be?
Why the CDF for the Normal Distribution can not be expressed as a closed form function? but does not provide any further details as to why this is the case, simply saying that the alternative is to write it in terms of the error function. Is there some way to intuit why the Normal Distribution can not be expressed as a closed form function?
When do you use the inverse CDF method?
The inverse CDF technique is particularly useful when you want to generate data from a truncated distribution. For a distribution F, if you generate uniform random variates on the interval [ F (a), F (b)] and then apply the inverse CDF, the resulting values follow the F distribution truncated to [ a, b ].
Is the CDF the same as a log?
There is indeed an historic convention of calling some functions “closed form”. However, once you note that most functions are actually computed using tables, or approximated using polynomials, then the CDF is not much different than a log. Thanks for contributing an answer to Cross Validated!
How to generate a random sample from a distribution?
If you know the cumulative distribution function (CDF) of a probability distribution, then you can always generate a random sample from that distribution. The inverse CDF technique for generating a random sample uses the fact that a continuous CDF, F, is a one-to-one mapping of the domain of the CDF into the interval (0,1).
Is the antiderivative of a Gaussian function closed?
The antiderivative of a Gaussian function has no closed form, but the integral over R can be solved for in closed form : ∫∞ − ∞exp( − x2)dx = √π. Since exp( − x2) is an even function (graph is symmetric about the y -axis), we can split this into two equal parts ∫∞ 0exp( − x2)dx = √π 2 = ∫0 − ∞exp( − x2)dx.
Which is an example of a closed form?
The gamma function is another such example. There is indeed an historic convention of calling some functions “closed form”. However, once you note that most functions are actually computed using tables, or approximated using polynomials, then the CDF is not much different than a log.