(a) The following experiments involving tossing a coin were conducted where the probability of obtaining a head is p. The intention is to estimate the parameter p using maximum likelihood principle....

Extracted text: (a) The following experiments involving tossing a coin were conducted where the probability of obtaining a head is p. The intention is to estimate the parameter p using maximum likelihood principle. 1) You tossed the coin 12 times and observed 8 heads and three tails. 2) Your friend tossed the coin until she observed the fourth head and noted that it occurred on the 10th toss. (i) Obtain an expression for the likelihood function in terms of p and calculate the MLE of p separately for both cases. Assuming the experiments in (1) and (2) are independent, calculate the joint (ii) likelihood function for them and obtain the MLE of p. Which of the three estimates of p would be more accurate and why? State clearly your reason and give evidence. (b) If gene frequency is in equilibrium, the genotypes AA, Aa and aa occur in a population with probabilities (1- 0)?, 20(1- 0), and 0², respectively where 0s0s1. A scientist recorded the genotypes of a sample of 190 people and the data are given below. Genotypes AA Aa aa Frequency 10 68 112 Obtain the likelihood of this grouped data and find the MLE Ô of 0. (i) (ii) Calculate the standard error of the MLE and obtain an approximate 95% confidence interval for 0. Based on your fitted model, calculate the expected frequencies of the three genotypes and comment on your results.