Ivo Dinov
UCLA Statistics, Neurology, LONI

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STAT 13 (1a, 1b, 1c)

Introduction to Statistical Methods for the Life and Health Science

Department of Statistics

## Instructor: Ivo Dinov

Homework 3
Due Date: Monday, Oct. 20, 2008

Please, submit your homework right before lecture on the due date. See the HW submission rules. On the front page include the following header.
• (HW_3) Solve the following  problems:
• Suppose that a disease is inherited via a sex-linked mode of inheritance, so that a male offspring has a 50% chance of inheriting the disease whereas the female offspring have no chance of getting the disease. Assume that 51.3% of births are male.
• In  a family with one male and one female offsprings, what is the probability that both sibling are affected?
• What is the probability that exactly one sibling is affected?
• What is the probability that neither sibling is affected?
• Suppose that a medical test has a 92% chance of detecting a disease if the person has it (i.e. 92% sensitivity) and a 94% chance of correctly indicating that the disease is absent if the person really does not have the disease (i.e. 94% specificity). Suppose that 10% of the population has the disease.
• What is the probability that a randomly chosen person will test positive?
• Suppose that a randomly chosen person does test positive. What is the probability that the person does have the disease?
• In a certain population of the European starling, there are 5,000 nests with young. The distribution of brood size (number of young in a nest) is given in the accompanying table. Suppose one of the 5,000 broods is chosen at random and let Y be the size of the brood. Find
• P(Y=3)
• P(Y>=7)
• P(4<=Y<=6)
• Calculate the mean of the random variable Y.
 Broad Size Brood Number 1 90 2 230 3 610 4 1400 5 1760 6 750 7 130 8 26 9 3 10 1 Total 5,000
• Consider a population of the fruit fly Drosophila melanogaster in which 30% of the individuals are black because of a mutation, while 70% of the individuals have the normal gray color. Suppose three flies are chosen at random from the population; let Y denote the number of black flies out of the three. Then the probability distribution for Y is given by the following table:
• Find Pr {Y ≥ 2}.
• Find Pr {Y ≤ 2}.
• Calculate the mean of Y.

 Y (No. Black) Probability 0 0.343 1 0.441 2 0.189 3 0.027 Total 1.00
• Go to the Coin Die Experiment.
• Simulate event independence between the outcome of the die (event B) and the outcome of the coin (event A), by setting the probabilities of both dice to be identical. Run 100 experiments and argue that the observed data implies independence between the events A={Coin=Head} and B={Die=3}, i.e., P(AB) = P(A) P(B), approximately.
• Now make the probability distributions of the two dice different (by clicking on the dice and manually changing the die probabilities). Show empirically the dependence of the probabilities, A={Coin=Head} and B={Die=3}.

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