Download EE 342 Mid-Term Name:

EE 342 Mid-Term
Closed Book, 1 crib sheet
Name:
Problem Maximum
Number Points
1
20
2
20
3
30
4
30
Total
100
Your
Score
1. (20 points) Suppose X is a Gaussian random variable with a zero mean and unit variance. Let
2X .
Y = e
(a) Express the cdf of Y in terms of the FX (x).
(b) Determine the pdf for Y .
2. (20 points) The two parts are independent.
(a) From the letters a; b; c; d; e and f , how many 4 letter codewords may be formed if
(i) no letter may be repeated
(ii) any letter may be repeated any number of times.
(b) Determine the number of possible integer solutions to the equation x1 + x2 + + x8 = 1000 where
each xi 0.
3. (30 points) Let K be the time (months) it would take for any particular bulb to burn out, i.e. K = 1
means a bulb burns out in the rst month. Suppose K is geometrically distributed with a mean 10
months. Four bulbs are tested simultaneously. Determine the probability of
(a) None of the 4 bulbs fails during its rst month of use.
(b) Exactly two bulbs have failed by the end of the second month.
(c) Exactly one bulb fails during each of the rst three months.
(d) Exactly one bulb has failed by the end of the second month, and exactly two bulbs are still working
at the start of the fth month.
4. (30 points) A man has 5 coins of which two are two-headed, one is two-tailed, and two are normal. He
does a few things sequentially. First, he closes his eyes and choses a coin at random and tosses it. Dene
the following events. Let H 2, T 2 and N be the respective events that the chosen coin is two-headed,
two-tailed or normal. Let Hu and Tu be the respective events that the tossed coin shows heads or tails
(u for up). Similarly, let Hd and Td be the respective events that the down side of tossed coin is heads or
tails. Note that if the chosen coin is two-headed, then both Hu and Hd occur.
(a) (i) Draw a probability tree to represent this scenario.
(ii) Determine the probability that the down side (not showing) of the coin is a head.
Second, the man then opens his eyes and sees that the coin is showing heads.
(b) Determine the probability that the down side of the coin is a head.
Third, the closes his eyes again and tosses the coin again.
(c) Suppose the coin shows heads again. Determine the probability that the chosen coin was the twoheaded coin.
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