Final answer:
The maximum likelihood estimation of X given a sample is the value of X that maximizes the likelihood function, which is 1/(b-a) for each observation in the sample. For the given sample x1 = 0.9, x2 = 0.2, x3 = 0.8, x4 = 0.1, the maximum likelihood estimation of a is 0.1 and b is 0.9. The Bayesian estimation of X given a sample can be calculated using a non-informative prior, which results in a posterior distribution that is the same as the maximum likelihood estimation.
Step-by-step explanation:
The maximum likelihood estimation of X given a sample is the value of X that maximizes the likelihood function. In this case, since X has a uniform distribution U([a, b]), the likelihood function is equal to 1/(b-a) for each observation in the sample.
To find the maximum likelihood estimation, we need to find the values of a and b that maximize this likelihood function.
For the given sample x1 = 0.9, x2 = 0.2, x3 = 0.8, x4 = 0.1, the maximum likelihood estimation of a is 0.1 and b is 0.9. These values maximize the likelihood function 1/(0.9-0.1) = 1/0.8 = 1.25 for each observation.
The Bayesian estimation of X given a sample involves taking into account prior knowledge or beliefs about the distribution of X.
Without any specific prior information, we can use a non-informative prior, which is a uniform distribution on the parameter space [0, 1].
This means that the prior probability of any value of X in the interval [0, 1] is the same.
Using Bayesian estimation with a non-informative prior, the posterior distribution of X given the sample is also a uniform distribution U([0.1, 0.9]).
This means that the Bayesian estimation of X is the same as the maximum likelihood estimation in this case.