1 Random Variables: Averaging polls Polling rms do lots of statistical modeling to improve the quality of their estimates. As in most modeling problems, they face a bias / variance tradeoff: the harder they try to adjust for all the different biases in their data, the more they will inate the variance. Suppose there are n polling rms each carrying out their own poll to estimate who intend to vote for the Republican candidate in a certain election. We therefore observe n random variables, X1, . . . ,Xn. Each X, is a result of the ith rm's poll indicating the proportion of voters who intend to vote for the Republican candidate, expressed in percentage points (i.e. 0 S X,- g 100 for all X,). The true percentage is some number 6, with 0 S 6 g 100. Due to factors like the sample size of each poll, the non-response bias, and the particularities of each rm's polling methodology, the random variable X,- has mean u, and variance 0?. Assume that the polls are independent. The true percentage is (9, which may or may not be equal to any of the ,u, values. (a) (2 points) Find the bias and MSE of X,, as an estimator of 6, in terms of the parame- ters \"\"03, and 6. (b) (2 points) Suppose we decide to aggregate the polls by taking a simple average: X 1 Z X Find the bias, variance, and MSE of X\" in terms of the parameters m and of for 2': 1,...,n, and 6. (c) (2 points) Generally speaking, we should expect the bias to matter more than the variance once we get a lot of polls, because the bias from different studies will not necessarily cancel out, but the random noise will tend to average out to zero over many studies. Suppose that we could either choose a world (World 1) where all of the polls have a bias of two percentage points in the same direction (all u,- = 6 + 2) and standard deviation of one percentage point (a,2 : 12), or a world (World 2) where all of the polls have a bias of 1 percentage point and standard deviation of ten percentage points. If we just want Xn to have as small MSE as possible, how large does n have to be for us to prefer World 2? (d) (2 points) Some polls may be higher quality than others, so we might want to give them more weight. Assume instead we are going to calculate a weighted average A I i wiXi, i:1 where the weights sum up to 1 (217% : 1). What are the bias, variance, and MSE of A