The likelihood function for the first five tosses X_1, ..., X_5 is given by:
L(X_1, ..., X_5; p) = p^3*(1-p)^2
The parameters of the Beta distribution equivalent to this likelihood function are:
α = 3 (the number of successes or heads obtained)
β = 2 (the number of failures or tails obtained)
The likelihood function for the first ten tosses X_1, ..., X_10 is:
L(X_1, ..., X_10; p) = p^7*(1-p)^3
The parameters of the Beta distribution for this likelihood function are:
α = 7 (the total number of successes obtained in the first ten tosses)
β = 3 (the total number of failures obtained in the first ten tosses)
Using the result from the previous problem, the mode of the Beta distribution is given by (α-1)/(α+β-2). Therefore, the MLE (Maximum Likelihood Estimation) in the frequentist view is (7-1)/(7+3-2) = 6/8 = 0.75.
Suppose that you have a coin with unknown probability p of landing heads; assume that coin toss outcomes are i.i.d Bernoulli random varaiables. You flip it 5 times and it lands heads thrice. Our parameter of interest is p. Compute the likelihood function for the first five tosses X_1, \ldots, X_5.
L(X_1, \ldots , X_5; p) =
The function you computed above is equivalent up to a constant of proportionality to a Beta distribution over 0 \leq p \leq 1. What are its parameters?
\alpha =
\beta =
Suppose that you flip it 5 more times and in the next five tosses it lands heads four times. Compute the likelihood function for the first ten tosses X_1, \ldots, X_{10}.
L(X_1, \ldots , X_{10}; p) =
Again, the function you computed above is equivalent up to a constant of proportionality to a Beta distribution over 0 \leq p \leq 1. What are its parameters?
\alpha =
\beta =
Using your result from the previous problem (“Mode of the Beta Distribution"), what is the MLE in the frequentist view?
1 answer
1 answer