Contents
How to write the likelihood of a Bernoulli?
Step one of MLE is to write the likelihood of a Bernoulli as a function that we can maximize. Since a Bernoulli is a discrete distribution, the likelihood is the probability mass function. The probability mass function of a Bernoulli X can be written as f(X) = pX(1 p)1 X.
Is the mean and variance of the Bernoulli distribution unknown?
Recall that the mean and variance of the Bernoulli distribution are E ( X) = p and var ( X) = p ( 1 − p). Often in statistical applications, p is unknown and must be estimated from sample data. In this section, we will see how to construct interval estimates for the parameter from sample data.
Which is an example of estimation in the Bernoulli model?
Estimation in the Bernoulli Model Recall that an indicator variable is a random variable that just takes the values 0 and 1. In applications, an indicator variable indicates which of two complementary events in a random experiment has occurred. Typical examples include
When is X a squence of Bernoulli trials?
That is, X is a squence of Bernoulli trials. From the examples in the introduction above, note that often the underlying experiment is to sample at random from a dichotomous population. When the sampling is with replacement, X really is a sequence of Bernoulli trials.
What is the variance of a Bernoulli distribution?
The variance of a Bernoulli distributed is [] = = We first find [] = (=) + (=) = + = = []
What is the MLE for repeated Bernoulli trials?
For repeated Bernoulli trials, the MLE p ^ is the sample proportion of successes. Suppose that X is an observation from a binomial distribution, X ∼ Bin ( n, p ), where n is known and p is to be estimated. The likelihood function is
Which is the maximum likelihood of a mle?
For the simple probability models we have seen thus far, however, explicit formulas for MLE’s are available and are given next. If our experiment is a single Bernoulli trial and we observe X = 1 (success) then the likelihood function is \\ ( L ( p ; x) = p\\). This function reaches its maximum at \\ (\\hat {p}=1\\).