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B02 學期總成績：
全班 54 人，
A 19， B 17， C
A+
A
AB+
B
BC+
C
CF
8， F 7 人， 退選 3 人
3
12
4
8
5
4
2
2
4
7
B02 屆
10/8/2014 quiz 1
1.
pm 9:10~ 10:00 新 302
(可使用計算機)
If X is a discrete random variable, how to describe it completely? If Y is a continuous
random variable, how to describe it completely? Use the appropriate formulas. (12%)
2. X is a random variable. Use the appropriate formulas to define its (a) sample skewness and (b)
sample coefficient of variation of. What are the their significant meaning? (12%)
3. Use the appropriate formulas to define (a) independence between two events A and B, and (b)
independence between two random variables X and Y. (12%)
4. The PDF of a random variable X is given as
f ( x) = k x2 -1＜ x ＜ 2
=0
elsewhere
Determine (a) P( 0＜ X ＜2 )
5.
f (x, y) = 6x
0 ＜ y ＜ 1,
(10%)
(b) P( X = 0.5) (4%)
0 ＜ x+y ＜ 1
=0
elsewhere
(a)Determine the marginal PDF of X and Y. (15%)
(b)Determine P(X＜0.2). (5%)
(c)Determine P( X ＞ 0.2∣Y = 0.5). (10%)
(d)Determine P( Y ≧ 0.6∣X = 0.5).
(10%)
(e)Determine whether X and Y are independent. (10%)
Engineering Statistics
Mid-Term
2004.11.12
1. The joint probability function of two random variables, X and Y, f(x,y) is
x
1
2
3
1
0.05
0.05
0.10
y
2
0.05
0.10
0.35
3
0.
0.20
0.10
a. Determine marginal distributions of X and Y.
b. Determine the first moments of X and Y.
c. Determine the covariance of X and Y.
d. Are X and Y statistically independent? (沒理由不計分) (20%)
2. Three machines A, B, and C, make 30%, 45%, and 25%, respectively, of the products. It is
known that 2%, 3%, and 4% of the products made by each machine, respectively, are defective.
a. What is the probability that a randomly selected product is defective? (5%)
b. If the randomly selected product is found to be defective, what is the probability that it is
made by machine B? (5%)
3. The joint density function of two random variables X and Y is
f ( x , y) = 2 0< x < y 0 < y < 1
= 0 elsewhere.
a. Determine if X and Y are independent.
b. Find Pr (X > 1/2).
c. Find Pr (X > 1/2 | Y = 2/3).
d. Find Pr (X > 1/2 | Y = 1/4).
(20%)
4. The return period of an earthquake with magnitude 7 in Central Taiwan is 100 years.
a. Determine the probability that this region will have two or three earthquakes with this
magnitude in the next 50 years (design life of a new structure).
b. If the service life of an old structure is 30 years, what is the probability that this structure
will experience the earthquake of magnitude 7? (20%)
5. A manufacturer of concrete claims that the strength of his product is, on the average, 4000 psi
with a standard deviation of 300 psi. Assume that the strength follows a normal distribution.
If six samples are selected and tested, determine the probability that this sample mean is
between 3900 psi and 4200 psi. (10%)
6. The tensile strength of a certain metal component is normally distributed with a mean 100 ksi
and a standard deviation of 3 ksi. Measurements are recorded to the nearest 0.5 ksi.
a. What is the proportion of these components exceeding 102 ksi in tensile strength?
b. Find the probability that among next 1000 testing samples, at least 250 samples will have
the tensile strength exceeding 102 ksi (20%).
12/10/2014 quiz 2 pm 9:10~ 10:00 新 302
開書測驗 (全部課本題目)
Engineering Statistics
Final Exam
2015.1.13
a.m. 10:20 ~ 12:10
Use of two columns on each page is suggested.
Page 1
on the answer sheet:
A. If it is found that 20 of 200 women in the city have breast cancer and 10 of 150 women in the
rural community have breast cancer. Can you conclude at  = 0.05 that breast cancer is more
prevalent in the city?
(20%)
B. The following data were recorded on sorbic acid residuals immediately after being dipped in a
sorbate solution and after 60 days storage:
Slice Before Storage
1
224
2
270
3
400
4
444
5
590
After Storage
116
96
239
329
437
6
660
597
7
1400
689
8
680
576
Assuming the populations to be normally distributed and using a 0.05 level of significance,
test the hypothesis that the length of storage does not influence sorbic acid residuals. (15%)
Page 2 &3
A. The fitted regression line for  Y
x
=    x is

y = a + b x, in which a  y  bx
 n
  n

b    ( xi  x )( yi  y )  /   ( xi  x ) 2  . In the analysis, it is assumed that  Y2i   2 = constant.
 i 1
  i 1

Note that  B2 =  2 / Sxx in which Sxx =  ( xi  x ) 2 .
Show that a 100(1-  )% confidence interval for the mean response  Y

y0  t / 2 s
2
1 ( x0  x )

 Y
n
S xx

x0
 y0  t / 2 s
2
1 ( x0  x )
.

n
S xx
x0
is
(12%)
B. The data of a structural test are as follows:
strain, x
1.2
1.4
1.7
stress, y
8.5
9.5
10.2
 xi = 8.2  yi = 47.9  xi2 = 13.9
1.9
2.0
9.2
10.5
 yi2 =461.43 and  xi yi = 79.32
(1) Carry out a regression analysis of Y on x. Estimate the stress when x = 1.8.
(14%).
(2) Evaluate the corresponding variance and the coefficient of determination.
(12%).
C. Based on the observed data points { ( xi , yi ) i  1 ~ N }, use the least square method to obtain

the fitted regression line y = a + c x 2 as an approximation of the model  Y | x     x 2 .
In short, determine A, B, C, D, E, and F in the following equation.
A
C

(12%)
B  a   E 

D  c   F 
Page 4
The probability function of random variable X is
f ( x)  e   x / x !
x = 0, 1, 2, …….
Suppose that a random sample 5, 6, 2, 4, 3, 6 is obtained from the distribution. Use the maximum
likelihood method to evaluate the estimate of  . (15%)
B01 屆
10/2/2013 quiz pm 9:10~ 10.05
1.
2
新 302
(可使用計算機)
Use two ways and the corresponding formulas to describe two discrete random variables X
and Y. (10%)
(a) Give an example of two mutually exclusive event. (5%)
(b) Use two formulas to define the independence of two events, A and B (5%)
3. What are the prior probability and the posterior probability? Use an example to illustrate it.
(10%)
4. Use an example to illustrate the sample mean and the population mean? (10%)
5. f (x, y) = 6y
=0
0 ＜ y ＜ x ＜ 1
elsewhere
(a) Determine the marginal distributions of X and Y. (10%)
(b) Calculate Pr ( Y ＜0.5∣ X = 0.7) (10%)
(c) Are X and Y independent? (5%)
(d) Calculate Pr ( 1 ＜ X+Y ) (15%)
6. Three tosses of a coin are performed. Let X be a random variable giving the number of heads
minus the number of tails in these trials.
Determine f(x). (20%)
Engineering Statistics
Mid Exam
2013.11.6
AM 10:20 ~ 12:10
Page 1 on the answer sheet:
The joint density function of two random variables X and Y is
f ( x , y) = c x y 0  x  y  2
= 0
elsewhere.
(a) Determine the value of c and Pr (X  1 | Y = 1.5 ). (10%)
(b) Determine Pr ( X +Y  2).
(10%)
Page 2
X is a normal variable, whose PDF is given as f ( x ) 
1
2 
exp( 
( x  7)2
).
4
There is a random variable Y = 3X + 6.
(a) Which random variable, X or Y, has a higher variability?
(8%)
(b) Determine the mean square of X. (4%)
(c) Determine the expectation of XY and the covariance of X and Y. (8%)
(d) Are X and Y uncorrelated? Are X and Y independent? Need an explanation. (5%)
Page 3
(a) On the average, in every 100 years, Taiwan has a severe earthquake with magnitude over 7.
Determine the probability that Taiwan will have more than 2 earthquakes with this
magnitude in the next 50 years, the design life of a new building. (10%)
(b) If the service life of an old building is only 20 years, what is the probability of the building
will experience such an earthquake? (6%)
(c) What are a Chi-Squared distribution and a t-distribution? (5%)
(d) What is the distribution of a sample mean? Answer this for different cases. (10%)
Page 4
A medicine company knows that 10% of its birth-control pills have an ingredient that is below
the minimum strength, thus rendering the pill ineffective.
(a) Calculate the probability that fewer than 2 in a sample of 20 pills will be ineffective (6%).
(b) Use a Poisson distribution to approximate (a). (6%).
(c) Use a normal distribution to approximate (a). (6%)
(d) Comment the results of (b) and (c). Which one is better?
Engineering Statistics
Final Exam
2014.1.7
(6%).
a.m. 10:20 ~ 12:10
Use of two columns on each page is suggested.
Page 1 on the answer sheet:
A. Six samples of a new material are tested to determine its strength index, which is a normal
random variable. The result is as follows: 17, 32, 14, 23, 26 and 17.
Determine 90% confidence intervals for its variance. (10%)
B. The fitted regression line for  Y
a  y  bx
x
=    x is

y = a + b x, in which
 n
  n

b   ( xi  x)( yi  y )  /   ( xi  x) 2  . In the analysis, it is assumed that
 i 1
  i 1

 Y2  constant =  2 . Denote A = Y – B x . Sxx =  ( xi  x ) 2 , Syy =  ( yi  y ) .
2
i
 B2 =  2 / Sxx.
Show that 
2
A
=
x
2
i
nS xx
 2 . (10%)
Page 2&3
The testing results concerning stress (x) and on the shear resistance (y) are as follows:
x
26.8, 25.4, 28.9, 23.6, 27.7, 23.9, 24.7, 28.1, 26.9, 27.4, 22.6, 25.6
y
26.5, 27.3, 24.2, 27.1, 23.6, 25.9, 26.3, 22.5, 21.7, 21.4, 25.8, 24.9
To perform a simple linear regression analysis, the following calculation is made,  xi = 311.6,
 yi = 297.2,  xi2 = 8134.26, Sxx = 43.0467, Syy = 47.1467 and Sxy = -29.5333.
(a) Estimate this regression line,  Y
x
=    x . (10%)
(b) Calculate the coefficient of determination. (5%)
(c) Calculate an estimate of  2 . (10%)
(d) Determine a 90% confidence interval for  . (10 %)
(e) Is this regression model a good one? (5%)
Page 3
It is argued that the resistance of wire A is greater than the resistance of wire B. An experiment on
the wires shows the following results (in ohms):
Wire A 127
Wire B 122
125
127
130
123
128
129
131
125
124
127
Assume equal variances. Perform a statistical hypothesis by establishing a critical region. What
conclusions can you draw from this? Use  = 0.05. (20%)
Page 4
To determine the distribution of the strength index of a particular material, we test 28 specimens.
Based on those data, we obtain a sample mean of 35 and a sample variance of 100.
The observed frequency is summarized as：
Class boundaries
observed number
10 ~ 20
2
20 ~ 30
4
30 ~ 40
9
40 ~ 50
7
50 ~ 60
6
Perform a goodness-of-fit test to determine if this strength index is a normal distribution,
using a 0.025 level of significance. (20%)