# Normal Distribution MLE

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Hello,

If I take, {X1, . . . , Xn}, a random sample from N(µ, σ

But, how do I do it with 2: the problem I get it 2 things:

E.g., suppose I am maximising the likelihood function of the population mean. Then, to find the MLE - usually, I just take the derivative of the likelihood function, or the log likelihood function l(µ) with respect to µ and then equate it to 0 to find the MLE of µ. This I mean if the only population parameter was µ.

But here, we have 2 parameters: so e.g. if we solve for the MLE of µ when σ

What I can tell is: We just find the likelihood function of L(µ) in terms of µ and σ

So, now I have got the likelihood function of µ which is the same for when σ

Thank you!

If I take, {X1, . . . , Xn}, a random sample from N(µ, σ

^{2}). So, this means I take a random sample of n objects from a population which is normally distributed with population mean µ and population variance σ^{2}. Now, I understand how to use the Maximum Likelihood Estimator (MLE) for distributions with 1 unknown parameter.But, how do I do it with 2: the problem I get it 2 things:

E.g., suppose I am maximising the likelihood function of the population mean. Then, to find the MLE - usually, I just take the derivative of the likelihood function, or the log likelihood function l(µ) with respect to µ and then equate it to 0 to find the MLE of µ. This I mean if the only population parameter was µ.

But here, we have 2 parameters: so e.g. if we solve for the MLE of µ when σ

^{2}is known, what do we do? And the same for if σ^{2}is unknown, what is the difference in the method?What I can tell is: We just find the likelihood function of L(µ) in terms of µ and σ

^{2}and then I would find the log likelihood function i.e. l(µ) = Log(L(µ)) with respect to base e i.e. natural logarithm.So, now I have got the likelihood function of µ which is the same for when σ

^{2}is known and unknown. What do I do after? When I differentiate with respect to µ, do I take partial derivatives or do something else? What is the difference in the method for the derivative part if σ^{2}is known or unknown?Thank you!

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#2

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**Chittesh14**)...

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Can't recall MLE myself, however it is fully worked, for the Normal distribution, in wiki - here.

**ghostwalker**)Can't recall MLE myself, however it is fully worked, for the Normal distribution, in wiki - here.

^{2}is known and for the other σ

^{2}is unknown, in both cases finding the MLE for µ and I can't distinguish the methods because the part where they differentiate is skipped and it is identical until then!

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#4

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Thank you, this is literally what is in my lecture notes haha, like almost identical. But, the only difference is in my lecture notes, it says for one: σ

**Chittesh14**)Thank you, this is literally what is in my lecture notes haha, like almost identical. But, the only difference is in my lecture notes, it says for one: σ

^{2}is known and for the other σ^{2}is unknown, in both cases finding the MLE for µ and I can't distinguish the methods because the part where they differentiate is skipped and it is identical until then!If is unknown, then the additional work of partial wrt is required, but not otherwise.

Or is it the actual differentating itself that's causing the problem?

Last edited by ghostwalker; 1 year ago

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(Original post by

As far as finding is concerned, I don't think there is any difference, since is held constant for the partial derivative.

If is unknown, then the additional work of partial wrt [tex]\sigma]/tex] is required, but not otherwise.

Or is it the actual differentating itself that's causing the problem?

**ghostwalker**)As far as finding is concerned, I don't think there is any difference, since is held constant for the partial derivative.

If is unknown, then the additional work of partial wrt [tex]\sigma]/tex] is required, but not otherwise.

Or is it the actual differentating itself that's causing the problem?

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Gregorius Tagged you in case you'd remember this topic.

So, my thoughts after reading the question I was stuck on again were:

If I take, {X1, . . . , Xn}, a random sample from the population with a distribution N(µ, σ

Is the following correct?

- If σ

- If σ

Thank you!

**If you just read this post rather than the first one, it is easier because my question is summarised in this post.**So, my thoughts after reading the question I was stuck on again were:

If I take, {X1, . . . , Xn}, a random sample from the population with a distribution N(µ, σ

^{2}). So, this means I take a random sample of n objects from a population which is normally distributed with population mean µ and population variance σ^{2}. Then, if I am estimating the population mean µ using the MLE (maximum likelihood estimator suppose), I do the normal steps and find the (log) likelihood function. Now, I have two cases: either the population variance σ^{2}is known or unknown.Is the following correct?

- If σ

^{2}is known, I simply do the normal method, take the partial derivative of the (log) likelihood function with respect to µ and set it equal to 0 and then solve for , labelling it the MLE of µ, which may be in terms of σ^{2}.- If σ

^{2}is unknown, then I have to use the normal method and take the partial derivative of the (log) likelihood function with respect to σ^{2}and set it equal to 0 and then solve for , labelling it the MLE of σ^{2}. Then, I find the normal MLE of µ as in the above step, and substitute this estimate for σ^{2}wherever σ^{2}is in the equation.Thank you!

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#7

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Gregorius Tagged you in case you'd remember this topic.

So, my thoughts after reading the question I was stuck on again were:

If I take, {X1, . . . , Xn}, a random sample from the population with a distribution N(µ, σ

**Chittesh14**)Gregorius Tagged you in case you'd remember this topic.

**If you just read this post rather than the first one, it is easier because my question is summarised in this post.**So, my thoughts after reading the question I was stuck on again were:

If I take, {X1, . . . , Xn}, a random sample from the population with a distribution N(µ, σ

^{2}). So, this means I take a random sample of n objects from a population which is normally distributed with population mean µ and population variance σ^{2}. Then, if I am estimating the population mean µ using the MLE (maximum likelihood estimator suppose), I do the normal steps and find the (log) likelihood function. Now, I have two cases: either the population variance σ^{2}is known or unknown.
- If σ

^{2}is unknown, then I have to use the normal method and take the partial derivative of the (log) likelihood function with respect to σ^{2}and set it equal to 0 and then solve for , labelling it the MLE of σ^{2}. Then, I find the normal MLE of µ as in the above step, and substitute this estimate for σ^{2}wherever σ^{2}is in the equation.
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(Original post by

So far, so good. Note that if the population variance is unknown, then the log-likelihood is a function of two variables, and . If the population variance is known, then it is a function of just one variable, latex]\mu[/latex]. So in the former case, to find the MLE, you're going to be finding the maximum of a function of two variables, which involves taking the partial derivatives, setting them equal to zero, and solving the consequent system of two equations in two unknowns. In the latter case, you are maximizing a function of one variable.

It's not a partial derivative in this case, as the likelihood is the function of just one variable, but otherwise correct.

This works in this case (i.e. for the normal distribution), as it is particularly simple. But in general, you should think in terms of setting both partial derivatives equal to zero (simultaneously) and solving the system of two simultaneous equations in two unknowns.

**Gregorius**)So far, so good. Note that if the population variance is unknown, then the log-likelihood is a function of two variables, and . If the population variance is known, then it is a function of just one variable, latex]\mu[/latex]. So in the former case, to find the MLE, you're going to be finding the maximum of a function of two variables, which involves taking the partial derivatives, setting them equal to zero, and solving the consequent system of two equations in two unknowns. In the latter case, you are maximizing a function of one variable.

It's not a partial derivative in this case, as the likelihood is the function of just one variable, but otherwise correct.

This works in this case (i.e. for the normal distribution), as it is particularly simple. But in general, you should think in terms of setting both partial derivatives equal to zero (simultaneously) and solving the system of two simultaneous equations in two unknowns.

Last edited by Chittesh14; 1 year ago

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#9

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Thank you so much, I get it completely now. You're too good ! So, in the case of where is known, I just differentiate the log-likelihood function with respect to normally treating as a constant right?

**Chittesh14**)Thank you so much, I get it completely now. You're too good ! So, in the case of where is known, I just differentiate the log-likelihood function with respect to normally treating as a constant right?

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