Question

Which pdfs would have the following moment-generating functions? a) Mᵧ (t)=e³ᵗ² b)Mᵧ (t)=4/(4−t) c)Mᵧ (t)=(1/4+3/4eᵗ)⁴

Answer 1

a) The PDF that would have the moment-generating function
`Mᵧ(t) = e^(3t^2)` is a Gaussian distribution.

b) The PDF that would have the moment-generating function Mᵧ(t) = 4/(4 - t) is an exponential distribution.

c) The PDF that would have the moment-generating function
`Mᵧ(t) = (1/4 + 3/4e^t)^4` is a binomial distribution.

Explanation:

The moment-generating function (MGF) of a probability distribution uniquely identifies the distribution. For Mᵧ(t) =
`e^(3t^2),` the MGF matches that of a Gaussian distribution, which is a characteristic of the normal distribution. The MGF of a normal distribution with mean μ and variance σ^2 is e^(μt + (σ^2t^2)/2). In this case, the MGF resembles e^(3t^2), indicating a mean of 0 and variance of 3/2, aligning with the standard normal distribution.

For Mᵧ(t) = 4/(4 - t), this MGF corresponds to an exponential distribution. The MGF of an exponential distribution with rate parameter λ is λ / (λ - t), and here, it aligns with the given expression when λ = 4.

Regarding Mᵧ(t) =
`(1/4 + 3/4e^t)^4,` this MGF resembles that of a binomial distribution. The MGF of a binomial distribution with parameters n and p is
`(1 - p + pe^t)^n.` Comparing the given MGF with this formula reveals its similarity, where n = 4 and p = 3/4.

Understanding the MGF helps in identifying the probability distribution associated with it, as each distribution has a unique MGF characteristic to it.