Math Problem Statement
Suppose Y is a random variable with cumulant generating function (cgf)
K Y ( t )
e t − 1.
What is the distribution of Y ?
N(0,1)
exponential (1)
Poisson(1)
U(0,1)
Solution
The cumulant generating function (CGF) is related to the distribution of a random variable by the fact that the cumulants (which are related to moments) can be derived from it.
The cumulant generating function is given as:
This form matches the CGF of a Poisson(1) distribution, because:
- For a Poisson(λ) distribution, the CGF is .
- When , the CGF simplifies to , which is exactly the given function.
Thus, the distribution of is Poisson(1).
Answer:
Poisson(1)
Would you like more details or have any questions?
Here are 5 related questions:
- What is the cumulant generating function for a normal distribution?
- How are cumulants related to moments in probability theory?
- What are the first four cumulants for the Poisson distribution?
- How can the CGF be used to find the mean and variance of a distribution?
- What is the CGF of an exponential distribution?
Tip: The cumulant generating function is especially useful because the derivatives of the CGF at give cumulants directly, simplifying calculations for higher-order moments!
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Cumulant Generating Function
Poisson Distribution
Formulas
K_Y(t) = e^t − 1
K_Y(t) for Poisson(λ) = λ(e^t − 1)
Theorems
Poisson Distribution CGF Theorem
Suitable Grade Level
Undergraduate Probability and Statistics
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