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Prоblem 1 (25 pts) A fаir die is rоlled twice. Let X{"versiоn":"1.1","mаth":"X"} be the sum of the vаlues rolled and Y{"version":"1.1","math":"Y"} the first value rolled minus the second value rolled. Find the correlation of X and Y. Are X and Y uncorrelated? Problem 2 (25 pts) A random variable X is conditionally exponential with a random parameter Λ{"version":"1.1","math":"Λ"}. This means that there is a random variable Λ such that X is exponential with parameter λ{"version":"1.1","math":"λ"} given that Λ = λ. Find the conditional distribution of Λ given X = x{"version":"1.1","math":"x"} if Λ is a geometric random variable with (non-random) parameter p{"version":"1.1","math":"p"}. You may leave your answer in terms of an infinite sum. Λ is a Rayleigh random variable with (non-random) parameter σ{"version":"1.1","math":"σ"}. You may leave your answer in terms of an integral. Note that your final answers should not be left in terms of the marginal density function of X. Problem 3 (25 pts) When a random current I{"version":"1.1","math":"I"} (in amperes) flows through a random resistance R{"version":"1.1","math":"R"} (in ohms), the power generated is Y=I2R{"version":"1.1","math":"Y=I2R"} (in watts). Suppose that I and R are independent random variables with probability density functions fI(x)=6x(1-x), 0≤x≤1{"version":"1.1","math":"fI(x)=6x(1-x), 0≤x≤1"} and fR(x)=2x, 0≤x≤1{"version":"1.1","math":"fR(x)=2x, 0≤x≤1"} Find the cumulative distribution function of Y{"version":"1.1","math":"Y"}. You may leave your answer in integral form. Problem 4 (25 pts) Consider a discrete-time random process Xn{"version":"1.1","math":"Xn"}, where Xn is a binary random variable for n=0, 1, .... {"version":"1.1","math":"n=0, 1, .... "} If P(X0=0)=p{"version":"1.1","math":"P(X0=0)=p"} and the process has the property that P(Xn=k Xn-1=j)=pjk{"version":"1.1","math":"P(Xn=k Xn-1=j)=pjk"}, for every n≥1{"version":"1.1","math":"n≥1"}, find the joint probability mass function of X1{"version":"1.1","math":"X1"} and X2{"version":"1.1","math":"X2"}. Note that probabilities p, p00, p01, p10, {"version":"1.1","math":"p, p00, p01, p10, "}and p11{"version":"1.1","math":"p11"} are parameters of the random process. Congratulations, you are almost done with the final exam. 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: Final Exam 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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