Solution Found!
Let X1,..., Xn be a random sample from a N(µ, ?2)
Chapter 4, Problem 10E(choose chapter or problem)
Let \(X_{1}, \ldots, X_{n}\) be a random sample from a \(N\left(\mu, \sigma^{2}\right)\) population. Find the MLEs of μ and of σ. (Hint: The likelihood function is a function of two parameters, μ and σ. Compute partial derivatives with respect to μ and σ and set them equal to 0 to find the values \(\widehat{\mu}\) and σ that maximize the likelihood function.)
Equation transcription:
Text transcription:
X_{1}, \ldots, X_{n}
N\left(\mu, \sigma^{2}\right)
\widehat{\mu}
Questions & Answers
QUESTION:
Let \(X_{1}, \ldots, X_{n}\) be a random sample from a \(N\left(\mu, \sigma^{2}\right)\) population. Find the MLEs of μ and of σ. (Hint: The likelihood function is a function of two parameters, μ and σ. Compute partial derivatives with respect to μ and σ and set them equal to 0 to find the values \(\widehat{\mu}\) and σ that maximize the likelihood function.)
Equation transcription:
Text transcription:
X_{1}, \ldots, X_{n}
N\left(\mu, \sigma^{2}\right)
\widehat{\mu}
ANSWER:
Solution :
Step 1 of 1:
Let be a random sample from a normal distribution with mean
and
.
The density for X is
f(x)=
Our goal is:
We need to find the maximum likelihood estimator of mean and variance
.
Now we have to find the maximum likelihood estimator of mean and variance
.
Here the likelihood function for the sample is
L(,
)=
=