(2pts * 8) Suppose there is a population with 20 short people and 60 tall people. You randomly sample one person. If the sampled person is short, record X=0 and then flip a coin. Then, if the coin comes up heads, record Y=0; if the coin comes up tails, record Y=1. If the sampled person is tall, record X=1 and Y=0. (That is, if the person selected is tall, you do not flip a coin.)

What is the joint p.m.f. of X and Y?
What is the marginal p.m.f. of X?
What is the conditional p.m.f. of Y given X = 0?
What is the conditional p.m.f. of Y given X = 1?
What is the conditional expectation function of Y given X?
What is the covariance of Y and X?
What is the (marginal) variance of X?
What is the best linear predictor of Y given X?
(1pt bonus) plot CEF and BLP using R.

(3pts) Why is the estimate of the standard error of the sample mean biased (Page 18, Section 9)?

(6pts) You are interested in estimating θ=μ_1-μ_2, where Y_1~Norm(μ_1,50) and Y_2~Norm(μ_2,100) and Y_1,Y_2 are independent. You can afford a total of 100 observations. How many observations should you draw from Y_1 and Y_2 respectively to ensure the variance of  θ ̂ is the lowest? Based on the estimator you propose, write the 95% confidence interval of θ ̂.

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