Problem 1 (25 pts) Let A{“version”:”1.1″,”math”:”A”} and B{“…
Problem 1 (25 pts) Let A{“version”:”1.1″,”math”:”A”} and B{“version”:”1.1″,”math”:”B”} be two events in a probability space (S, F, P){“version”:”1.1″,”math”:”(S, F, P)”}. Is it true that P(A\B)=P(A)-P(B){“version”:”1.1″,”math”:”P(A\B)=P(A)-P(B)”}? Why or why not? As a reminder, A\B={ω∈S: ω∈A, ω∉B}{“version”:”1.1″,”math”:”A\B={ω∈S: ω∈A, ω∉B}”}. Problem 2 (25 pts) Consider a random experiment with sample space S{“version”:”1.1″,”math”:”S”} = {1, 2, 3, 4, 5, 6}, where the outcomes in S{“version”:”1.1″,”math”:”S”} are all equally likely. Let A{“version”:”1.1″,”math”:”A”} and B{“version”:”1.1″,”math”:”B”} be independent events. Assuming that A{“version”:”1.1″,”math”:”A”} has four elements, what values can B{“version”:”1.1″,”math”:”B”} take, where B{“version”:”1.1″,”math”:”B”} is the number of elements in B{“version”:”1.1″,”math”:”B”}? Problem 3 (25 pts) Consider two probability measures P1{“version”:”1.1″,”math”:”P1″} and P2{“version”:”1.1″,”math”:”P2″} defined on the same event space F{“version”:”1.1″,”math”:”F”}. Under what conditions on the real-valued constants a1{“version”:”1.1″,”math”:”a1″} and a2{“version”:”1.1″,”math”:”a2″} is the function a1P1(A)+a2P2(A), ∀A∈F{“version”:”1.1″,”math”:”a1P1(A)+a2P2(A), ∀A∈F”} , a valid probability measure? Problem 4 (25 pts) A coin is tossed with P(H)=P(T)=12{“version”:”1.1″,”math”:”P(H)=P(T)=12″}. If the coin comes up heads, you lose 2 dollars. If it comes up tails, you are equally likely to win any amount of money in (0, 10) dollars, i.e., any real number between 0 and 10 dollars. Let X{“version”:”1.1″,”math”:”X”} be the amount of money you win. Find the cumulative distribution function of X. Note that X is negative if you lose money. Congratulations, you are almost done with Exam 1. 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 1 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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