Final answer:
The moment generating functions for different distributions are explained.
Step-by-step explanation:
a) Beta Distribution: The moment generating function for the beta distribution does not have a closed form expression. Therefore, it is not possible to identify it based on the moment generating function.
b) Uniform Distribution: The moment generating function for the uniform distribution on the interval [a, b] is given by M(t) = (e^(bt) - e^(at)) / (t(b - a)).
c) Exponential Distribution: The moment generating function for the exponential distribution with rate parameter λ is given by M(t) = 1 / (1 - t/λ).
d) Gamma Distribution: The moment generating function for the gamma distribution with shape parameter k and rate parameter θ is given by M(t) = (1 - t/θ)^(-k).
e) Lognormal Distribution: The moment generating function for the lognormal distribution with parameters μ and σ is not available in closed form. Therefore, it is not possible to identify it based on the moment generating function.
f) Normal Distribution: The moment generating function for the normal distribution with mean μ and variance σ^2 is given by M(t) = e^(μt + (σ^2t^2)/2).
g) Weibull Distribution: The moment generating function for the Weibull distribution with shape parameter k and scale parameter λ is given by M(t) = (1 - t/λ)^(-k).