Let x1, x2, , xn be a random sample from n(μ, σ2), where the mean θ = μ is such that −[infinity] < θ < [infinity] and σ2 is a known positive number. show that the maximum likelihood estimator for θ is θ^ = x.

And then the maximum occurs when , and that is only satisfied if and only if:

Step-by-step explanation:

For this case we have a random sample where where is fixed. And we want to show that the maximum likehood estimator for .

The first step is obtain the probability distribution function for the random variable X. For this case each have the following density function:

The likehood function is given by:

Assuming independence between the random sample, and replacing the density function we have this:

Taking the natural log on btoh sides we got:

Now if we take the derivate respect we will see this:

And then the maximum occurs when , and that is only satisfied if and only if:

the 3rd option is right

step-by-step explanation: