This applet shows the (log of the) likelihood function, and the maximum likelihood estimator, for a sample and various statistical models.

This applet demonstrates the principle of maximum likelihood estimation. The blue curve represents a possible population generating the data, with parameter [color=#1551b5]θ[/color]. You can change population types by using the buttons at top-right. An actual observed sample [color=#000000][math]x_1, x_2,[/math][/color] ... is shown along the horizontal axis. You can adjust the sample size n using the slider at the top-left. You can move individual sample observations by dragging them. The likelihood of each [color=#000000][math]x_i[/math][/color] for the current [color=#1551b5]θ[/color] value is shown as a dotted blue line. You can drag the blue point to adjust [color=#1551b5]θ[/color], and observe how the likelihoods of the sample points are affected. You can also key in a new value for [color=#1551b5]θ[/color] in the text box. Try to guess the MLE by dragging [color=#1551b5]θ[/color] and observing the likelihoods, particularly for sample sizes 1 and 2. You can show the likelihood of the whole sample, and the MLE, using the 'Likelihood function' checkbox. You can switch to the log-likelihood of the whole sample using the 'Log-likelihood' checkbox. You can also use the zoom buttons and arrows to adjust the views of the windows.