*11*

A uniform distribution is a probability distribution in which every value between an interval from *a *to *b *is equally likely to be chosen.

The probability that we will obtain a value between x_{1} and x_{2} on an interval from *a *to *b *can be found using the formula:

P(obtain value between x_{1} and x_{2}) = (x_{2} – x_{1}) / (b – a)

This tutorial explains how to find the **maximum likelihood estimate (mle)** for parameters *a *and *b *of the uniform distribution.

**Maximum Likelihood Estimation**

**Step 1: Write the likelihood function.**

For a uniform distribution, the likelihood function can be written as:

**Step 2: Write the log-likelihood function. **

**Step 3: Find the values for a and b that maximize the log-likelihood by taking the derivative of the log-likelihood function with respect to a and b.**

The derivative of the log-likelihood function with respect to *a *can be written as:

Similarly, the derivative of the log-likelihood function with respect to *b *can be written as:

**Step 4: Identify the maximum likelihood estimators for a and b.**

Notice that the derivative with respect to *a *is monotonically increasing. Thus, the mle for *a* would be the largest *a *possible, which would simply be:

min(X_{1}, X_{2}, … , X_{n})

Also notice that the derivative with respect to *b *is monotonically decreasing. Thus, the mle for *b *would be the smallest *b *possible, which would be:

max(X_{1}, X_{2}, … , X_{n})