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?
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.
To compute the likelihood function for the first five tosses X_1, ..., X_5, we can use the fact that the toss outcomes are independent and identically distributed (i.i.d) Bernoulli random variables.
The likelihood function is given by:
L(X_1, ..., X_5; p) = P(X_1 = x_1, ..., X_5 = x_5; p) = P(X_1 = x_1; p) * P(X_2 = x_2; p) * ... * P(X_5 = x_5; p)
Since each toss has two possible outcomes (heads or tails), the probability of landing heads is p and the probability of landing tails is 1-p. Therefore, we can express each probability as:
P(X_i = x_i; p) = p^(x_i) * (1-p)^(1-x_i)
Substituting this expression into the likelihood function, we get:
L(X_1, ..., X_5; p) = p^(x_1) * (1-p)^(1-x_1) * p^(x_2) * (1-p)^(1-x_2) * ... * p^(x_5) * (1-p)^(1-x_5)
Simplifying further, we can write the likelihood function as:
L(X_1, ..., X_5; p) = p^(x_1 + x_2 + ... + x_5) * (1-p)^(5 - x_1 - x_2 - ... - x_5)
Therefore, the likelihood function for the first five tosses is:
L(X_1, ..., X_5; p) = p^3 * (1-p)^(5-3)