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How do I get the MGF PDF?
In general you can show this by converting the MGF to the characteristic function ΨX(ω), i.e. replacing t by jω, and then use the inverse fourier transform to get the pdf.
How do you find the distribution of a moment-generating function?
The mgf MX(t) of random variable X uniquely determines the probability distribution of X. In other words, if random variables X and Y have the same mgf, MX(t)=MY(t), then X and Y have the same probability distribution.
What is a moment-generating function used for?
Not only can a moment-generating function be used to find moments of a random variable, it can also be used to identify which probability mass function a random variable follows.
What is PDF and PMF in probability?
Probability mass functions (pmf) are used to describe discrete probability distributions. While probability density functions (pdf) are used to describe continuous probability distributions.
Why does the moment generating function not converge?
It may be that the moment generating function does not exist, because some of the moments may be infinite (or may not have a definite value, due to integrability issues). Also, even if the moments are all finite and have definite values, the generating function may not converge for any value of t other than 0.
Is it easy to invert a generating function?
Inverting generating functions is not always easy. It does help to state any constraints on the parameters (which are left unstated here by the OP). Anyway, the approach is: First, convert your mgf into a characteristic function (i.e. replace t -> I t):
Which is the moment of the exponential distribution?
Moments of the exponential distribution. We know from Exam-ple 6.1.2 that the mgf mY(t) of the exponential E(t)-distribution is 1 1 tt. It is not hard to expand this into a power series because 1 1 tt is nothing by the sum of a geometric series 1 1 tt = ¥ å k=0 tktk. It follows immediately that m k = k!tk. Last Updated: September 25, 2019