Final answer:
The confusion in the question stems from mixing normal and uniform distributions and unrelated concepts from the Central Limit Theorem. For a normal distribution, the MLEs for μ and σ2 do exist but cannot be calculated without sample data. The question inaccurately implies MLEs do not exist for these parameters.
Step-by-step explanation:
The question seems to be confused as it refers to both a general normal distribution X ~ N(μ, σ2) and a uniform distribution X ~ U(0,1), which are different statistical distributions. For a normal distribution, the maximum likelihood estimators (MLE) of μ and σ2 do exist and are given by the sample mean and variance, respectively, when sample data is provided. However, without sample data, we cannot calculate MLEs. For the uniform distribution X ~ U(0,1), the parameters μ and σ are clearly defined as 0.5 and √((1-0)2/12) respectively. The question contains several unrelated fragments that refer to the Central Limit Theorem (CLT) which is not directly connected to the existence of MLEs for a given distribution. The CLT pertains to the distribution of sample means and not the parameter estimation for a population.