What is the moment generating function (MGF) of the Poisson distribution?

What is the moment generating function (MGF) of the Poisson distribution?

• A. Mx(t) = e^(λ((e^t)-1))
• B. Mx(t) = (1 - λe^t)^-1
• C. Mx(t) = λe^t
• D. Mx(t) = e^(λt)

Answer: (A)

The moment generating function (MGF) of a Poisson distribution is derived from the definition of the MGF, which is given by the expected value of \( e^{tX} \) where \( X \) is a random variable representing the Poisson process. The Poisson distribution is characterized by a single parameter \( \lambda \), which indicates the average rate (or mean) of occurrences in a fixed interval of time or space. To find the MGF, we start with the definition: \[ M_X(t) = E[e^{tX}] = \sum_{k=0}^{\infty} e^{tk} \cdot P(X = k) \] For a Poisson random variable, the probability mass function is \( P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!} \). Substituting this into our MGF formulation gives us: \[ M_X(t) = \sum_{k=0}^{\infty} e^{tk} \cdot \frac{e^{-\lambda}\lambda^k}{k!} \] This can be rearranged: \[ M_X(t) = e^{-\lambda} \sum

The moment generating function (MGF) of a Poisson distribution is derived from the definition of the MGF, which is given by the expected value of ( e^{tX} ) where ( X ) is a random variable representing the Poisson process. The Poisson distribution is characterized by a single parameter ( \lambda ), which indicates the average rate (or mean) of occurrences in a fixed interval of time or space.

To find the MGF, we start with the definition:

[

M_X(t) = E[e^{tX}] = \sum_{k=0}^{\infty} e^{tk} \cdot P(X = k)

]

For a Poisson random variable, the probability mass function is ( P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!} ). Substituting this into our MGF formulation gives us:

[

M_X(t) = \sum_{k=0}^{\infty} e^{tk} \cdot \frac{e^{-\lambda}\lambda^k}{k!}

]

This can be rearranged:

[

M_X(t) = e^{-\lambda} \sum