Suppose we have a sample of 15 independent and identically distributed observations X1, X2, ..., X15 from a normal distribution with an unknown mean μ and known variance σ²=4. We want to estimate the value of μ using the method of maximum likelihood.
The probability density function of a normal distribution with known variance is given by:
f(x|μ, σ²) = (1/√(2πσ²)) * exp[-(x-μ)²/(2σ²)]
where x is the observed value of the random variable X.
What is the likelihood function for this sample, and what is the maximum likelihood estimate of μ?
Data: 4,3,7,8,3,6,7,5,4,6,7,8,5,6,9