Assume Z and X are independent random variables with Z~N(0,1) and X~Bernoulli(p),
where 0 < p < 1 . Define Y = X if Z > 0 and Y = −X if Z ≤ 0 .
(1) Find the probability function of Y. (10%)
(2) Find the covariance of X and Y. (10%)
Automobiles arrive at a vehicle equipment inspection station according to a Poisson
distribution with mean λ = 5 every half hour. Suppose that with probability of 0.5 an
arriving vehicle will have no equipment violations.
(1) What is the probability that exactly 10 arrivals during an hour and all 10 have no
violations? (5%)
(2) For any fixed y ≥ 10 , what is the probability that y arrivals during an hour, of
which 10 have no violations? (6%)
(3) What is the probability that 10 "no-violation" cars arrive during the next hour? (6%)
A sample of 300 urban adults reveals 63 who favored increasing the highway speed
limit, whereas a sample of 180 rural adults yielded 75 who favored the increase.
(1) Use the data to test if the sentiment for increasing the speed limit is the same for the
two groups of adults? You should state the hypotheses first. (5%)
(2) Construct a 95% confidence interval for the difference of the proportions of the two
groups. (5%)
(3) If the true proportions favoring the increase are actually p1 = 0.2 (urban) and
p2 = 0.4 (rural), what is the probability that the "no-difference" hypothesis will be
rejected at α = 0.05 with the above information? (5%)
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