L20.10 Maximum Likelihood Estimation Examples  Summary and Q&A
TL;DR
Maximum likelihood estimation is a method used to estimate unknown parameters based on observed data.
Questions & Answers
Q: What is maximum likelihood estimation?
Maximum likelihood estimation is a statistical method used to estimate unknown parameters by maximizing the likelihood function, which is the probability of obtaining the observed data given the parameter values.
Q: How does maximum likelihood estimation work in the context of a binomial random variable?
In the case of a binomial random variable, the likelihood function is derived by calculating the probability of obtaining a specific number of successes (heads) in a series of coin flips. The maximum likelihood estimate for the parameter (probability of heads) is obtained by maximizing this likelihood function.
Q: Can maximum likelihood estimation be applied to normally distributed random variables?
Yes, maximum likelihood estimation can be applied to normally distributed random variables. In this case, the likelihood function is derived from the product of the probability density functions of the individual observations. The maximum likelihood estimates for the mean and variance are obtained by optimizing the likelihood function with respect to these parameters.
Q: How are the maximum likelihood estimates related to sample statistics?
In both examples, the maximum likelihood estimates coincide with commonly used sample statistics. In the case of the binomial random variable, the estimate for the probability of heads is equal to the observed proportion of heads. In the case of normally distributed random variables, the estimate for the mean is equal to the sample mean, and the estimate for the variance is a variation of the sample variance formula.
Summary & Key Takeaways

Maximum likelihood estimation is a procedure used to estimate unknown parameters in statistical models.

In the first example, a binomial random variable is used to demonstrate the maximum likelihood estimation process.

The second example involves estimating the mean and variance of a set of normally distributed random variables.