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IE 111 Fall 2009
Homework #6 Solutions
Question 1.
To check on the security screeners at an airport, we put 4 fake bombs into a set of 52
baggage pieces.
Suppose that the screeners actually just pick 6 pieces randomly (without
replacement) to inspect.
Let X be the number of our fake bombs that they find. Find
V(X)
V[X] = 0* P
X
(x=0) + 1* P
X
(x=1)+ 4* P
X
(x=2)+ 9* P
X
(x=3)+ 16* P
X
(x=4) – E[X]
2
= 0.384
Question 2.
The “coefficient of variation” is defined to be the standard deviation divided by the mean.
It is what we use when we say something like “cars weigh 2000 pounds, plus or minus
15%”—the 15% is the coefficient of variation.
Recall that the standard deviation is the
squareroot of the variance.
a)
Let V be a Binomial random variable with n=100,p=0.40; what is the coefficient
of variation?
CF = sqrt(V[X])/E[X] = sqrt(np(1p))/np = 0.122
b)
Let Y be a Pascal random variable with r=2, p=1/20.
What is its coefficient of
variation?
CF = sqrt(V[X])/E[X] = sqrt(r(1p)/p
2
)/(r/p) = sqrt((1p)/r) = 0.689
c)
Let Z be a Pascal random variable with r=4, p=1/10.
What is its coefficient of
variation?
CF = sqrt(V[X])/E[X]
= sqrt(r(1p)/p
2
)/(r/p) = sqrt((1p)/r) = 0.474
Question 3.
The total amount of snow during a winter in the Lehigh valley is a random variable with
mean 40 inches and variance 100 inches.
My son is wishing for 100 or more inches of
snow this winter.
What can you tell him about the probability of his wish?
μ
= 40
σ
= 10
If we choose k=6 then Chebyshev’s inequality becomes:
P(
μ
k
σ
< X <
μ
+k
σ
)
≥
1  (1/k
2
)
P(40 6*10 < X < 40+6*10)
≥
1  (1/6
2
)
P(20 < X < 100)
≥
0.97222
So his chances are less than 0.028
1
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On Tuesdays, patients arrive to the hospital at a rate of 2.5 per hour according to a
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This note was uploaded on 02/21/2010 for the course IE 111 taught by Professor Storer during the Spring '07 term at Lehigh University .
 Spring '07
 Storer

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