Problem 1. (10 points total) Suppose that X1, X2, . . ., Xm are i.i.d. continuous random variables with probability density function, or PDF, given by Base1 for 0 g :1: S 1, f(x|0) = { 0 otherwise. Here, 6 6 [0,1] is an unknown parameter. 3:9 for 0 S x g 1, Equivalently, the cumulative distribution function, or GDP, is F(:.3|6) = P(Xz- g 2:) = O for a: 1. (a) (2/5 pts.) What is the Maximum Likelihood estimate 9m, of the parameter 6' (as a function of X1, X2, . . . , X\")? (b) (2.5 pts.) What is the approximate variance of the Maximum Likelihood estimate ML for large n? (c) (2.5 pts.) Suppose that n = 100 and ML = 1.5. Give an approximate 95% condence interval for 6. (d) (2.5 pts.) The median of the distribution of a continuous random variable X is a number m 6 IR for which P(X S m) = 1/2. Using ML from part (c), what is the estimate of the median of the distribution of each of the random variables Xi? Hint: use the expression given above for the CDF