Question

Problem 1. Consider independent tosses of the same coin. The probability that the coin will land on head is p and tails is 1-p. HHTTTTHHTHHHHHHHHTTHHHHHH

a. What is the probability of this specific series of tosses (also known as the likelihood of the data)? Assume each toss is independent. (2 pts)

b. Is the likelihood greater for p = .6 or p = .7? (1 pt)

c. What value of p maximizes the likelihood for n tosses of which k are H and n-k are T? (2 pt)

Answer 1

a) The probability that the coin will land on head is p and tails is 1-p. There are 18 H and 7 T in this data

HHTTTTHHTHHHHHHHHTTHHHHHH

ans:

The probability/likelihood of this specific series of tosses is

b) The likelihood for p=0.6 is

The likelihood for p=0.7 is

ans: the likelihood greater for p = 0.7

c) the likelihood for n tosses of which k are H and n-k are T is

The log of this likelihood is

To maximize l(p), we equate the first derivative of l(p) w.r.t p to 0 (the first order condition)

ans: The value of p maximizes the likelihood for n tosses of which k are H and n-k are T is k/n