Question
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?
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?
Answers
Answered by
GPT 3.5
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.
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.