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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 Qfunction 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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 Fall '14
 Central Limit Theorem, Probability theory, Alice, Geometric distribution

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