Let X be a standard normal random variable so that X∼N(0,1). For this problem you may want to refer to the table provided on Canvas. Recall, that Φ(x) denotes the standard normal CDF. (a) Find Φ(1.45). (b) Find x, such that Φ(x) = 0.4. (c) Based on the fact that Φ(1.645) = 0.95 find an interval in which X will fall with 95% probability. (d) Find another interval (different from the one in (c)) into which X will fall with 95% probability. Hint you may want to consider the symmetric properties of the standard normal distribution. (e) Find an interval with lower bound -1.9 into which X will fall with 95% probability. (f) Find an interval with upper bound 2.1 into which X will fall with 95% probability. (g) Among all possible intervals into which X falls with 95% probability, find the shortest one.

Imagine you have two estimators X and Y, for unknown parameter µ. Assume that they are estimated from independent datasets. Let Z = }(X+Y) be the average of these two estimators. To write the symbol µ, you can write mu. For other math notation, you can E[X], Var(X) and (X+Y)/2 or (1/2)(X+Y). Part a - If X and Y are unbiased estimators, E[X] = E[Y] = µ, then is Z unbiased? Explain why or why not. Your explanation should include a short derivation. Part bL Assume Var(X) = Var(Y). Is Z a lower variance or higher variance estimator than X? Explain your answer. Your explanation should include a short derivation.

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Hello, could I please get a solution for A through D