# MLE (Maximum Likelihood Estimate) for Normal Distribution

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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.