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1. [20 points] “Time headway” in traffic flow is the elapsed time between the time that one car
finishes passing a fixed point and the instant that the next car begins to pass that point.
Let X = time headway (in seconds) for two randomly chosen consecutive cars on a
freeway during a period of heavy flow. The pdf of X is given below. .15e− .15( x − .5 ) f (x) = 0 x ≥ .5
otherwise (i) [5 points] Find the cumulative distribution function (cdf ) of the time headway. (ii) [5 points] Use the cdf in part (i) to find the probability that the time headway is more
than 5 seconds. Is the median of the distribution of time headway more than 5
seconds? Explain your answer. (iii) [5 points] Find the expected value, E[X], of the random variable time headway.
[Hint: E[X0.5] may be found using a standard continuous distribution.] (iv) [5 points] Due to a terrorist warning, a traffic inspection team starts a quick spot
check of every vehicle at a location that is upstream of the fixed point at which the
headway is being recorded. The time taken (in seconds) to do a quick check of each
vehicle is a random variable Y, with Gamma (α = 1, β = 2) distribution. Find the
expected value of the time headway, X+Y, during the period when the spot check is
being done. 1 NAME__________________________
2. [20 points] A manufacturing operation runs 24 hours per day, 7 days a week. The accident
rate at this plant is λ = 2.1 accidents per week. The distribution of the number of accidents Xt at
this plant, during a period of time t, is a Poisson distribution with mean µ = λ∗ t.
a) [4 points] Find the probability, p, that there is at least one accident at this plant on a randomly selected day. Explain your answer. b) [4 points] Find the probability that there is at most one accident in any given week, i.e., P[ X 1 ≤ 1]. Explain your answer. c) [3 points] As a model for some characteristic of accidents during a week, the Safety Engineer uses the random variable Y with a Binomial distribution with n = 7 and the value of p given in
part (a) above. Explain what the Safety Engineer is modeling with the random variables X1 and
Y? d) [9 points] For 365 days of a year, let Y , Y2 , …, Y365 represent independent Bernoulli trials
1
denoting whether the day had ‘one or more accident’ or it was ‘accident free.’ Give the exact and
365 approximate distributions of ∑Y i , the number of days with one or more accidents, during the i =1 365 next year (365 days)? Find the approximate probability that ∑Y
i =1 2 i is between 90 and 110. NAME__________________________
3. [25 points] From a legislative committee consisting of Four Republicans, Three Democrats
and Two independents, a subcommittee of Three persons is to be randomly selected to
discuss possible compromises on an issue of importance. Let X = number of Democrats, and
Y = number of Republicans on this subcommittee. The joint pmf of (X, Y), p(x, y), with some
entries missing, is given in the following table:
p( x , y ) 0 1 0
x ? 1 3
84
6
84
1
84 2
3 y 24
84
12
84
0 p(y) 3 12
84
18
84 4
84 0 0 0 0 30
84 ? 2 4
84 0 p(x) 45
84
18
84
1
84 (i) [5 points] Fill in the missing probabilities (?) for the cells corresponding to
{X=0,Y=0}, {X=0, Y=1}, and explain your answers. (ii) [7 points] Calculate the marginal pmf of Y, and find the Expected number of
Republicans, E[Y], on the subcommittee. (iii) [8 points] Given that there is one Democrat on the subcommittee, find the conditional
pmf of the number of Republicans, Y, on the subcommittee. (iv) [5 points] Find the expected number of Republicans on the subcommittee, given that
there is exactly one Democrat on the subcommittee. 3 NAME__________________________
4. [15 points] Suppose the expected tensile strength of typeA steel is 105 ksi and the standard
deviation of tensile strength is 8 ksi. For the typeB steel, suppose the expected tensile
strength and the standard deviation of tensile strength is 100 ksi and 6 ksi, respectively. Let
X = the sample average tensile strength of a random sample of 32 typeA specimens, and let
Y = the sample average tensile strength of a random sample of 36 typeB specimens.
(a) [5 points] What is the approximate distribution of X ? Of Y ? Justify your answer. (b) [5 points] What is the approximate distribution of X  Y ? Justify your answer. (c) [5 points] Use part (b) to calculate (approximately) P[ 0 ≤ X  Y ≤ 2] . 4 NAME__________________________
5. [20 points] For the following problems, circle the correct answer from among the
multiple choices. For every correct answer you will earn 2.5 points . You should resist
guessing, since for every wrong answer, you will lose 1.5 points .
a) [2.5 points] If X and Y are independent random variables with distributions
p x (0) = .5, px (1) = .3, px ( 2) = .2 and p y (0) = .6, p y (1) = .1, p y ( 2) = .25, p y ( 3) = .05.
respectively. Then P[{ X ≤ 1} ∩ {Y ≤ 1} ] is
A.
B.
C.
D. .30
.56
.70
.80 b) [2.5 points] Suppose that the cumulative distribution function (cdf ) of a continuous
random variable X is 0,
x<0 2
F ( x) = 1 − (1 − x ) , 0 ≤ x < 1 1,
1≤ x The 75th percentile of the distribution of X is
(i)
(ii)
(iii)
(iv) 1 − (0.75) 1 /2
1 −− (0.5)1/2
0.5
0.75 c) [2.5 points] If X is a continuous random variable defined on the interval [ ,B], and the
A
probability density function of X is
1/( B − A)
f (x ; ,B ) = A
0 A≤ x≤ B
otherwise then X is said to have
A.
B.
C.
D. gamma distribution
normal distribution
uniform distribution
Lognormal distribution 5 NAME__________________________
d) [2.5 points] If ρ is the correlation coefficient between two variables X and Y, then which
of the following statements are false?
A.
B.
C.
D.
E. If X and Y are independent, then ρ = 0
If ρ = 0 , then X and Y are independent.
ρ = −1 or +1 if and only if Y = aX + b for some numbers a and b, with a ≠ 0
ρ is a measure of the degree of linear relationship between X and Y.
All of the above statements are false e) [2.5 points] Let X be a discrete random variable with V(X) = 8.6, then V(3X + 5.6) is
A.
B.
C.
D. 77.4
14.2
83.0
31.4 f) [2.5 points] The major difference between the binomial and hypergeometric distributions
is that with the hypergeometric distribution
A.
B.
C.
D.
E. the probability of success must exceed .5
the trials are independent of each other
the probability of success is not the same from trial to trial
N is large, but n is small
None of the above statements are true g) [2.5 points] The weekly rainfall amount (in inches) for a section of the midwestern
United States is known to follow a right skewed distribution. Which distribution should
be preferred most as a model for the weekly rainfall?
(i)
(ii)
(iii)
(iv) Binomial distribution
Normal distribution
Gamma distribution
Uniform distribution h) [2.5 points] Which dataset has the smallest standard deviation?
(i)
(ii)
(iii)
(iv) {2, 1, 0, 1, 2}
{0.001, 0.002, 0.003, 0.004}
{5, 10, 15, 20}
{1000, 1000, 1000, 1000, 1000} 6 ...
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 Spring '09
 Statistics, Normal Distribution, Probability theory, probability density function, ρ, Safety Engineer

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