Page 257 - Fundamentals of Probability and Statistics for Engineers

240 Fundamentals of Probability and Statistics for Engineers

7.4 Repeat Problem 7.3 if the distribution of the rod diameter remains uniform but
2
0:
that of the sleeve inside diameter is N(2 cm, 0004 cm ).
7.5 The first mention of the normal distribution was made in the work of de Moivre in
1733 as one method of approximating probabilities of a binomial distribution when
n is large. Show that this approximation is valid and give an example showing
results of this approximation.
7.6 If the distribution of temperature T of a given volume of gas is N(400, 1600),
measured in degrees Fahrenheit, find:
(a) f (450);
T
(b) P(T 450);
(c) P( T j m T j 20);
(d) P( T j m T j 20 T j 300).
7.7 If X is a random variable and distributed as N(m, 2 ), show that

1=2
2

EfjX mjg ˆ :

7.8 Let random variable X and Y be identically and normally distributed. Show that
random variables X ‡ Y and X Y are independent.

7.9 Suppose that the useful lives measured in hours of two electronic devices, say T 1
and T 2 , have distributions N(40, 36) and N(45, 9), respectively. If the electronic
device is to be used for a 45-hour period, which is to be preferred? Which is
preferred if it is to be used for a 48-hour period?
7.10 Verify Equation (7.13) for normal random variables.
7.11 Let random variables X 1 , X 2 ,... , X n be jointly normal with zero means. Show that
E X 1 X 2 X 3 gˆ ; 0
f
EfX 1 X 2 X 3 X 4 gˆ EfX 1 X 2 gEfX 3 X 4 g‡ EfX 1 X 3 gEfX 2 X 4 g‡ EfX 1 X 4 gEfX 2 X 3 g:

Generalize the results above and verify Equation (7.35).
7.12 Two rods, for which the lengths are independently, identically, and normally
distributed random variables with means 4 inches and variances 0.02 square inches,
are placed end to end.
(a) What is the distribution of the total length?
(b) What is the probability that the total length will be between 7.9 inches and 8.1
inches?
7.13 Let random variables X 1 , X 2 , and X 3 be independent and distributed according
to N(0, 1), N(1, 1), and N(2, 1), respectively. Determine probability P(X 1 ‡ X 2 ‡
X 3 > 1).
7.14 A rope with 100 strands supports a weight of 2100 pounds. If the breaking strength
of each strand is random, with mean equal to 20 pounds and standard deviation 4
pounds, and if the breaking strength of the rope is the sum of the independent
breaking strengths of its strands, determine the probability that the rope will not
fail under the load. (Assume there is no individual strand breakage before rope
failure.)

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