Fn 17. Please answer
1. You buy one share of stock in company C for C10. Each day the price of C either increases by C1 with probability p or decreases by E1 with probability 1-p. These changes from day to day are statistically independent. You decide to sell your share if it gains C2 (i.e. reaches a price of (12). (1) What is the probability that you will sell your share exactly 4 days after you buy it ? [5 marks] (ii) What is the probability that you sell your share at least 4 days after you buy it ? [5 marks] Suppose now that the daily change in the price of stocks in company C is observed to be related to the change in price of stocks in company D. Namely, the probability that stock in C increases by E1 is equal to 0.2 when the price of stock in company D increases that day, and is equal to 0.1 otherwise. (ili) State the definition of conditional probability. [5 marks] (iv) Describe how marginalisation can be used to calculate the probability of an event E based on knowledge of the conditional probabilities P(E | F1), P(E| F2) and P(E | Fa) plus the probabilities P(F.), P(F2) and P(Fa) when events F1 , Fa, Fa are mutually exclusive and Fi U F U F, equals the sample space. [5 marks] (v) Suppose that the probability that stock in company D increases on a given day is 0.5. Calculate the probability that stock in company C increases that day. [5 marks] 2. Suppose you play a game where four 6-sided fair dice are rolled. Let X be equal to the minimum of the four values rolled (it is ok if more than one dice has the minimal value). It costs $2 to play the game and you win EX. (i) Calculate P(X2k) as a function of k=1,2,...,6. [5 marks] (ii) Assuming you know P(X2k) for k=1,2,..,6, show how to calculate the PMF of X. [5 marks] State the definition of the expected value. [5 marks] (iv) Calculate E[X]. [5 marks] (v) If you play the game many times do you expect to make a profit (win more than you pay to play the game) ? Explain your reasoning. What is the amount cost to play that would make you break even (i.e. have an expected profit of zero) ? [5 marks]3. A survey is carried out by selecting n people from the population and asking each person to answer either "yes" or "no" to a question. Let random variable Y, take value 1 when the i'th respondent answers "yes" and 0 otherwise. The random variables Y, i=1,2,..,n are independent and identically distributed with E[Y ]= H. (i) Let random variable Z = [i=" Y. Write an expression for E[Z] in terms of E[Yi]. Explain your answer. Hint: use the linearity of the expected value. [5 marks] (ii) Using the definition of expectation prove that E[Z]=E[Z] for n>0. [5 marks] (iii) Using Chebyshev's inequality explain the weak law of large numbers and the behaviour of |Z - u | as n becomes large. Recall that for random variable X Chebyshev's inequality is: P(|X - u|2k) s E[(X- u)?]/k? for any k and u. [5 marks] (iv) Explain what a confidence interval is, using Z as an estimate of u as an example. [5 marks] ( v ) Describe how to use bootstrapping to estimate a confidence interval for Z. [5 marks] 4. Suppose we mark the answers of 200 students to each of 10 exam questions. Let Sy be an indicator variable which is 1 if student i answered question j correctly and -1 otherwise. You observe all of the answers for all students. Assume that P(Sy | a, dj) = 1/(1+exp(-y(a-dj)) where a is a parameter that represents the students ability and dj is a parameter which represents the questions difficulty. (i) Give an expression for the log-likelihood of this exam data (the data consisting of the answers by all 200 students). Hint: this is an example of a logistic regression model. [5 marks] (ii) Outline how gradient descent might be used to find the maximum likelihood estimates for the unknown parameters a, and dj. [5 marks] (iii) With reference to Bayes Rule explain what is meant by the likelihood, prior and posterior. [5 marks] (iv) Explain how the maximum a posteriori (MAP) estimate of a parameter differs from the maximum likelihood estimate. [5 marks] (v) How could you incorporate knowledge of the prior probability distribution of parameters a into the above model to obtain a MAP estimate ? [5 marks]
