The Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a probability distribution that best fit the observed data. In the context of a normal distribution, MLE is used to estimate the population mean (μ) and standard deviation (σ) from a sample of data.
Likelihood Function: The likelihood function for a normal distribution is given by:
\[ L(\mu, \sigma) = \prod_{i=1}^{n} \left(\frac{1}{\sigma \sqrt{2\pi}}\right) \cdot e^{-\frac{(x_i - \mu)^2}{2\sigma^2}} \]
Sample Histogram
Parameter Comparison
Population Mean (μ)
0
Population Standard Deviation (σ)
1
Estimated Mean (MLE)
0
Estimated Standard Deviation (MLE)
1
Likelihood Function Contour Plots
Low Prob.High Prob.