In this article, we explain how to find the Maximum Likelihood Estimate (MLE) for the normal distribution to estimate the parameters µ and σ. The Maximum Likelihood Estimate is obtained by maximizing the value of the Likelihood Function.
Step 1: Write the Probability Distribution Function.
Let X be a random variable following the normal distribution with parameters µ and σ. The probability distribution function is given as,
Step 2: Find the Likelihood Function.
Suppose that X1, X2,…, and Xn are sample values extracted from a population following the normal distribution. We obtain the likelihood function by taking the product of the probability distribution of each Xi from i=1 to n.
Step 3: Find the Log Likelihood Function and compute Derivatives:
The Log Likelihood Function is obtained by taking the logarithm of the likelihood function obtained above. The Log Likelihood function is given as,
The derivative with respect to the parameter µ is given as,
The derivative with respect to the parameter σ is given as,
Step 4: Set the Derivatives equal to Zero
Since we want to maximize the log likelihood function we set the first derivative to 0 in order to find the critical points.
Setting the derivative with respect to µ equal to 0 we get,
Σ(Xi – µ) = 0
ΣXi – nµ = 0
µ = ΣXi / n = X̅
Therefore, the MLE (Maximum Likelihood Estimate) for the population mean is given by the sample mean.
Similarly setting the derivative with respect to σ equal to 0 we get,
Therefore, the MLE (Maximum Likelihood Estimate) for the population standard deviation is given by the standard deviation of the sample values.
