Problem 1 (25 pts) Let X{“version”:”1.1″,”math”:”X”} be the…
Problem 1 (25 pts) Let X{“version”:”1.1″,”math”:”X”} be the number of full pairs of dots observed in a toss of a fair die. (In other words, X=N/2{“version”:”1.1″,”math”:”X=N/2″}, where N{“version”:”1.1″,”math”:”N”} is the value rolled and x{“version”:”1.1″,”math”:”x”} is the largest integer less than or equal to x{“version”:”1.1″,”math”:”x”}.) Find the variance E[(X-E[X])2]{“version”:”1.1″,”math”:”E[(X-E[X])2]”}. Problem 2 (25 pts) Consider the function g{“version”:”1.1″,”math”:”g”} given by g(x)={“version”:”1.1″,”math”:”g(x)=”} {x2-1,x1){“version”:”1.1″,”math”:”P(Yln(1+X2)>1)”}. You may need the relation ∫11+x2dx=tan-1x+C{“version”:”1.1″,”math”:”∫11+x2dx=tan-1x+C”} for some constant C{“version”:”1.1″,”math”:”C”}. Problem 4 (25 pts) Let X{“version”:”1.1″,”math”:”X”} be a random variable uniformly distributed on [0, 1], and consider the floor function g(x)=[x]{“version”:”1.1″,”math”:”g(x)=[x]”}, where [x]{“version”:”1.1″,”math”:”[x]”} is the largest integer less than or equal to x{“version”:”1.1″,”math”:”x”}, for any real number x≥0{“version”:”1.1″,”math”:”x≥0″}. Now let Y=g(nX)+1{“version”:”1.1″,”math”:”Y=g(nX)+1″} for some fixed positive integer n{“version”:”1.1″,”math”:”n”}. Find the probability mass function of Y{“version”:”1.1″,”math”:”Y”}. Congratulations, you are almost done with Exam 2. DO NOT end the Honorlock session until you have submitted your work to Gradescope. When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to go to the assignment in Gradescope: Exam 2 Submit your answer sheets. Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.
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