in distribution?) that this 3. (10 points) Suppose X1, X2, ..., Xn are continuous random variables with a distribution, f (x; /), that has a mean of u. Suppose further that Y1, Y2, ..., Yn are the discrete random variables that result from rounding X1, X2, ..., Xn to the nearest integer. We can write Y; = Xi + 6;, where E; is uniformly distributed on [-=, =]. Show that Yn converges in probability to /. 4. (10 points) Let X1, X2, ..., Xn be independent, identically distributed normal random variables. Let $2 be the sample variance. a. Show that $2 is a consistent estimator for the population variance, o2. b. Is the sample standard deviation S = VS2 a consistent estimator for the population standard deviation, o? Why or why not? 5. (30 points) Let X1, X2, ..., Xnbe independent, identically distributed random variables with p.d.f. f(x) = 0x0-1, 0sx51. a. Discuss the regularity conditions in terms of the problem of estimating the value of 0 for this distribution. b. Find the maximum likelihood estimator, 0, for 0. c. Is 0 is a consistent estimator for 0