課程名稱︰工程統計學
課程性質︰必修
課程教師︰朱致遠
開課學院：工學院
開課系所︰土木工程學系
考試日期（年月日）︰102/11/15
考試時限（分鐘）：110分鐘
試題 :
Engineering Statistics Midterm Exam (2013/11/5)
● Exam time: 10:20am-12:10pm.
● Closed-book.
● Calculators are allowed. Cell phones are strictly prohibited.
● The calculation process must be described explicitly, including the
● probability functions for the commonly used distributions.
● 30% of the total score.
1. (3%) Suppose that the four inspectors at a film factory are supposed to
stamp the expiration date on each package of film at the end of the assembly
line. John, who stamps 20% of the packages, fails to stamp the expiration
date once in every 200 packages; Tom, who stamps 60% of the packages, fails
to stamp the expiration date once in every 100 packages; Jeff, who stamps
15% of the packages, fails to stamp the expiration date once in every 90
packages; and Pat, who stamps 5% of the packages, fails to stamp the
expiration date once in every 200 packages. If a customer complains that her
package of film does not show the expiration date, what is the probability
that it was inspected by John?
2. (3%)
(a) Given a standard normal distribution, find the area under the curve that
lies between z=-1.97 and z=0.86.
(b) Given a standard normal distribution, find the value of k such that
P(k<Z<-0.18)=0.4197.
(c) Given a random variable X having a normal distribution with μ=50 and
σ=10, find the probability that X assumes a value between 45 and 62.
3. (3%) Show that the mean of a random variable following a geometric
distribution is 1/p, where p is the probability of a success. The proof is
equivalent to showing that the return period (迴歸期) of an event occurring
with probability p is 1/p.
4. (3%) Let X, Y, and Z be independent variables where
E(X)=1, V(X)=3
E(Y)=4, V(Y)=7
E(Z)=3, V(Z)=2
What is the mean and variance of (a)U=3X+4Y (b) V=Y-3Z (c) W=U+V
5. (3%) Flaws occur in mylar material according to a Poisson distribution with
a mean of 0.01 flaw per square yard.
(a) If 25 square yards are inspected, what is the probability that there are
no flaws?
(b) What is the probability that a randomly selected square yard has no
flaws?
(c) Suppose that the material is cut into 10 pieces, each being 1 yard
square. What is the probability that 8 or more of the 10 pieces will have
no flaws?
6. (3%) Determine the mean and variance of X.
2
f(x)=kx for 0<x<4
7. (6%) Given the joint density function
2
12y , 0<x<1-y,0<y<1
f(x,y)={
0 elsewhere
Determine the following.
(a) If X and Y are independent.
(b) E(Y|X)
(c) P(Y>2|X=0.6)
8. (3%) It is assumed that 4000 of the 10,000 voting residents of a town are
against a new sales tax. If 15 eligible voters are selected at random and
asked their opinion, what is the probability that at most 7 favor the new
tax?
(a) Using hypergeometric distribution.
(b) Using binomial approximation.
9. (3%) Let X denote the number of bars of service on your cell phone whenever
you are at an intersection with the following probabilities:
┌────┬────┬────┬────┬────┬────┬────┐
│x │0 │1 │2 │3 │4 │5 │
├────┼────┼────┼────┼────┼────┼────┤
│P(X=x) │0.1 │0.15 │0.25 │0.25 │0.15 │0.1 │
└────┴────┴────┴────┴────┴────┴────┘
Determine the following:
(a) F(x)
(b) Mean and variance
(c) P(X<2)