Homework 4 Solution

# 04 of the actual probability with probability 095

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Unformatted text preview: : 1. . Suppose that we have graduates and get i.i.d. random variables 1 when the ith graduate attends the graduating ceremony, 0 when the We define ith graduate doesn’t attend the graduating ceremony. 1 and 0 1 . (1’) ∑ Denote the estimate as 0.04, . . (2’) From the Chebyshev inequality: | | 1 Solve the inequality, we get that 1 0.95 (1’) 3125. (1’) 4. (Lec. 21, 5 points) Packet transmission time on a certain Internet link is independent and identically distributed according to a geometric distribution with mean 3 ms. Suppose 800 packets are transmitted. Use the central limit theorem to estimate the required transmission time so that all the packets can be transmitted with probability 99%. (Hint: the Q-function table can be found on slides 27 in Lec. 11). Solution: Denote the transmission time for the ith packet is ms, 1. . , 800. 3, we get that For geometric variable: and 6. (1’) ∑ The total transmission time: ms According to the central limit theorem, approaches a Gaussian distribution with mean 2400 and variance 4800. (2’) Let denote the required transmission time, to satisfy the requirement: 1 0.99 (1’) √ After solving the inequality above, we get that 2561ms. (1’) 5. (Lec. 23, 5 points) We flip a fair coin and define a random process sin if the head shows, and 2 if the tail shows. (a) Find . (2 points) for 1. (3 points) (b) Find Solution: The random process can be expressed as sin 2, sin (a) (b) | (2’) 0, ,0 1, 0 2 (3) 2 , if head shows; if tail shows. as follows:...
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