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Let X have a Poisson distribution.
Therefore Pr(X=x) = e^(-k) * k^x/x! where x = 0, 1, 2, ...
Normally the parameter is the Greek letter, lambda, but this site does not allow any such symbols!
Then the moment generating function of the Poisson distribution is
E[exp(tx)] = Sum[e^(tx)*Pr(X=x)] where the sum is over all non-negative integers, x.
= Sum[e^tx * e^(-k) * k^x/x!]
= Sum[(e^t)^x * e^(-k) * k^x/x!]
= e^(-k) * Sum[(e^t)^x * k^x/x!]
= e^(-k) * Sum[(k*e^t)^x /x!]
= e^(-k) * e^(k*e^t)
= e^[k*(e^t - 1)]
What else can I help you with?
Obtaining moment generating function of poisson distribution?
The MGF is exp[lambda*(e^t - 1)].
How do you obtain the moment generating function of a Poisson distribution?
Using the Taylor series expansion of the exponential function. See related links
Which distribution function has equal mean and variance?
The exponential distribution and the Poisson distribution.
Moment generating and the cumulant generating function of poisson distribution?
The moment generating function is M(t) = Expected value of e^(xt) = SUM[e^(xt)f(x)] and for the Poisson distribution with mean a inf = SUM[e^(xt).a^x.e^(-a)/x!] x=0 inf = e^(-a).SUM[(ae^t)^x/x!] x=0 = e^(-a).e^(ae^t) = e^[a(e^t -1)]
What are the key considerations when implementing a C program that simulates a Poisson distribution?
When implementing a C program to simulate a Poisson distribution, key considerations include understanding the Poisson distribution formula, generating random numbers using a Poisson distribution, and ensuring the program accurately reflects the expected distribution outcomes. Additionally, it is important to validate the results of the simulation and optimize the program for efficiency.
Which distribution is used to find probabilities about the number of independent events occurring in a fixed time period with a known average rate?
The Poisson distribution. The Poisson distribution. The Poisson distribution. The Poisson distribution.
Is the Poisson probability distribution discrete or continuous?
The Poisson distribution is discrete.
If X has Poisson distribution does aX plus b have Poisson Distribution?
Yes.
Why belong exponential family for poisson distribution or geometric distribution?
Why belong exponential family for poisson distribution
How do you compute the probability distribution of a function of two Poisson random variables?
we compute it by using their differences
How do you calculate the mean and variance of a poisson distribution as a function of time t?
Divide the total number of incidents by the total time. The result, representing the average number of incidents per unit of time, is the mean as well as the variance of the Poisson distribution.
What is the difference between poisson distribution and poisson process?
A poisson process is a non-deterministic process where events occur continuously and independently of each other. An example of a poisson process is the radioactive decay of radionuclides. A poisson distribution is a discrete probability distribution that represents the probability of events (having a poisson process) occurring in a certain period of time.
