MATH&STAT 3460 Assignment&Ans 2 - MATH/STAT 3460...

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MATH/STAT 3460, Intermediate Statistical Theory Winter 2014 Toby Kenney Homework Sheet 2 Due: Friday 31st January: 3:30 PM Basic Questions 1. A certain genetic trait is controlled by a single gene with two versions A and B. Each person has two copies of the gene. There are therefore three possibilities: AA, AB, and BB. If the probability that a randomly chosen gene is A is p , the probabilities of the three possibilities should be p 2 , 2 p (1 - p ) and (1 - p ) 2 respectively. A sample of 100 people is taken and yields the following results: AA — 37 people, AB — 42 people, BB 21 people (a) Using a normal approximation, find a 5% likelihood interval for p . (b) Find a suitable power transformation of p to improve the accuracy of the normal approximation.
2. Let X 1 , X 2 , X 3 be independent samples from a normal distribution distri- bution with mean μ and variance 1, conditional on X i > 0 for all i . That is, X i has density function e - ( x - μ ) 2 2 2 π Φ( μ ) . Suppose X 1 + X 2 + X 3 = 5. Use New- ton’s method to find the maximum likelihood estimate for μ . [You only need to find the value to two decimal places. You will need to calculate Φ( μ ) for some values. You can look it up on a normal table.] 3. Let X 1 , X 2 , X 3 , X 4 be distributed as the sum of two exponential distri- butions with parameters λ 1 and λ 2 , with λ 1 < λ 2 . If the values of X 1 , X 2 , X 3 , X 4

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