Let Y denote a Bernoulli(θ) random variable with 0 < θ < 1. Suppose we are interested in estimating the odds ratio, γ = θ/(1 − θ), which is the probability of success over the probability of failure. Given a random sample {Y1,...,Yn}, we know that an unbiased and consistent estimator of θ is Y¯, the proportion of successes in n trials. A natural estimator of γ is G = Y¯ /(1−Y¯), the proportion of successes over the proportion of failures in the sample

a. Why is G not an unbiased estimator of γ (show this matematically and explain)?
b. Use the properties of plim (mathmatically) to show that G or Y/(1-Y) is a consistent estimator of γ.

Exclude leap years from the following calculations. (a) compute the probability that a randomly selected person does not have a birthday on october 4. (type an integer or a decimal rounded to three decimal places as needed.) (b) compute the probability that a randomly selected person does not have a birthday on the 1st day of a month. (type an integer or a decimal rounded to three decimal places as needed.) (c) compute the probability that a randomly selected person does not have a birthday on the 30th day of a month. (type an integer or a decimal rounded to three decimal places as needed.) (d) compute the probability that a randomly selected person was not born in january. (type an integer or a decimal rounded to three decimal places as needed.)