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10. a. Xis a Gaussian random variable with EMI = 0 and Var(X) = 6. Evaluate P (X 10). b. Y is a Gaussian random variable with E[Y] = -2 and Var(Y) = 9. Evaluate Pa YI 11). c. Z is a Gaussian random variable with E121 = 5 and Var(Z) = 4. What approximate region contains 60% of the most likely values of Z?
See less9. Yi and Y2 are independent continuous exponential random variables with A, and 22.X= max(YI,Y2). a. Find the CDF and PDF of X. b. Find the mean and variance of X.
See less8. A microcontroller can execute only one command at a time. Two commands are received 6 microseconds apart. The execution time for the first command is independent of the second and can be modeled as an exponential random variable with a mean of 5 microseconds. The execution of the second command can similarly be modeled as an exponential random variable with a mean of 10 microseconds. What is the expected time from the start of execution of the first command to the completion of the second command?
See less7. II is a continuous uniform random variable taking values from -1 to 3. Y2 is an independent uniform random variables taking values from 0 to 4. Z= max(Yj, Y2). H= min(Y,, Y2). a. Find the CDF and PDF of Z. b. Find the mean and variance of Z. c. Find the CDF and PDF of H. d. Find the mean and variance of H.
See less6. Suppose continuous random variable X is uniformly distributed over [-2, 3] and Y=X2. a. Find the CDF and the PDF of Y. b. Find E[Y].
See less5. A certain casino game costs $1 to play and has an outcome of a loss with probability p, which nets $0 (a loss of the $1), or an outcome of a win with probability 1-p, which nets $2 (winnings of $1 + $1 from upfront cost). Each game costs exactly $1. a. What is the probability that a player starting with $1 will end up losing all winnings on exactly the Th game? Use p = 0.3. b. What is the probability that a player starting with $1 will end up losing all winnings in less than 10 games? Use p = 0.5. c. Find the probability that a player starting with $1 will lose all of their winnings eventually, as a function p. Plot your answer.
See less4. The probability of a particular individual hitting the dart board in a particular throw is 0.8, independent of other throws. Z is the random variable representing the number of hits on target in 10 throws. a. Calculate and plot the PMF of Z. b. Calculate and plot CDF of Z. (You may desire to manually adjust the plot per convention) c. Using CDF values, find the probability that 3 < z < 6 throws were on target. d. Find E[Z] and var[Z]
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