The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Apr 14, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due Apr 20, 2015
Solution to Practice Problem 9
1. Since there are a large number of drivers, each of whom has a small probability of
being involved in an accident in a gi
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Apr 14, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due Apr 20, 2015
Practice Problem 9
1. The number of trac accidents in Berkeley, California, in 10 randomly chosen nonrainy days in 1998 is as follows: 4, 0, 6, 5, 2, 1, 2
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Apr 27, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due May 4, 2015
Solution to Practice Problem 11
1. By Testing the Means of Two Samples with known Variances
H0 : 1 2 , H1 : 1 > 2
X Y
TS =
2
x /n
+
5.6 4.1
=
22 /9 + 22
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Apr 27, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due May 4, 2015
Practice Problem 11
1. The value received at a certain messagereceiving station is equal to the value sent
plus a random error which is normal, with mean
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Mar 3, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due Mar 6, 2015
Practice Problem 4
1. X is a Binomial random variable with parameters (n, p).
(a) Find the moment generating function of X.
n k nk
)
(Hint: (a + b)n = n
k=0
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Apr 20, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due Apr 24, 2015
Solution to Practice Problem 10
1. (a)
x t0.005,8 s/ 9 = 1.0056 (3.355)(0.0246/3) = 1.0056 0.0275
The 99% condence interval on the mean diameter is given
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Mar 16, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due Mar 20, 2015
Practice Problem 6
1. First write out the Excel commands and then give the numerical answer for each
calculation.
(a) P cfw_
2
10
5 and
2
0.1,10
(b) P cf
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Mar 31, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due Apr 10, 2015
Practice Problem 8
1. A useful way of comparing two data sets is to put their stemandleaf plots side by
side. The following represents the scores of stu
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Apr 14, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due Apr 20, 2015
Practice Problem 9
1. The number of tra c accidents in Berkeley, California, in 10 randomly chosen nonrainy days in 1998 is as follows: 4, 0, 6, 5, 2, 1,
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Engineering Probability and Statistics
Jiheng Zhang
Solution to Practice Problem 1
1. (a) cfw_(i, j, k) : i = H or T, j = H or T, k = H or T .
(b) cfw_x : 0 x n, n is the total number of leaves on the plant, x is integer.
(c) [0, ).
(d) (0, ).
(
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Engineering Probability and Statistics
Jiheng Zhang
Practice Problem 2 Solutions
1. Dene A the event that exactly two heads occur. Dene Bi the event that the ith
outcome is head.
(a)
P (AB1 ) =
c
c
P (B1 B2 B3 ) + P (B1 B2 B3 )
0.5 0.5 0.5 + 0.
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned May 4, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due May 8, 2015
Practice Problem 12
1. (a) Sxx = 334.1, SY Y = 19.3, SxY = 66.9, SS = 5.9
(b) y = 0.2x + 5.7
(c)
H0 : = 0; H1 : = 0
(n 2)Sxx
B = 4.3
SS
t/2,n2 = t0.025,8 =
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Apr 27, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due May 4, 2015
Solution to Practice Problem 11
1. By Testing the Means of Two Samples with known Variances
H0 : 1 2 , H1 : 1 > 2
X Y
TS =
2
x /n
+
5.6 4.1
=
22 /9 + 22
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Mar 9, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due Mar 13, 2015
Practice Problem 5
1. Suppose that the length of a phone call in minutes is an exponential random variable
with parameter = 1/10. If someone arrives immedi
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Mar 31, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due Apr 10, 2015
Practice Problem 8
1. A useful way of comparing two data sets is to put their stemandleaf plots side by
side. The following represents the scores of stu
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Mar 23, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due Mar 26, 2015
Practice Problem 7
1. (a) For independent standard random variables X and Y , we have X + Y
N (0, 2). However if we further set Y = X which gives Y N (0,
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned May 5, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due May 8, 2015
Practice Problem 12
1. An electric utility wants to estimate the relationship between the daily summer temperature(degrees Fahrenheit) and the amount of ele
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Engineering Probability and Statistics
Jiheng Zhang
Solution to Practice Problem 4
1. (a)
M (t) = E[etX ]
n
etk
=
k=0
n
=
k=0
t
n k
p (1 p)nk
k
n
(pet )k (1 p)nk
k
= (pe + 1 p)n
(b) M (t) = n(pet + 1 p)n1 pet
M (t) = n(n 1)(pet + 1 p)n2 (pet )2
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Engineering Probability and Statistics
Jiheng Zhang
Solution to Practice Problem 5
1. Let X denote the length of the call made by the person in the booth. Then the
desired probabilities are
P (X > 10) = 1 FX (10) =
1
e
P (10 < X < 20) = FX (20)
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Mar 31, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due Apr 10, 2015
Solution to Practice Problem 7
1. (a) The summation of normal random variables to be a normal r.v., their is a
premise: independence. If we set Y = X, it
The Hong Kong University of Science and Technology
Engineering Probability and Statistics
IELM 2510

Spring 2015
IELM 2510
Assigned Mar 31, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due Apr 10, 2015
Solution to Practice Problem 8
1. (a) School A: xA = 83.708, xA = 87 and sA = 12.167.
School B: xB = 76.546, xB = 78 and sB = 12.459.
Since the sample mea
The Hong Kong University of Science and Technology
probability
IELM 2510

Spring 2016
IELM 2510
Assigned Feb 23, 2016
Engineering Probability and Statistics
Jiheng Zhang
Due Feb 29, 2016
Practice Problem 3
1. Roll two fair dice. Let X denote the result of the outcome of the first dice subtract
the outcome of the second dice.
(a) What are t
The Hong Kong University of Science and Technology
probability
IELM 2510

Spring 2016
IELM 2510
Assigned Mar 8, 2016
Engineering Probability and Statistics
Jiheng Zhang
Due Mar 14, 2016
Practice Problem Solution 5
1. To find the CDF, we need to integrate the PDF, with a dummy variable (x is used
here), from the beginning of the support (x
The Hong Kong University of Science and Technology
probability
IELM 2510

Spring 2016
IELM 2510
Assigned Apr 5, 2016
Engineering Probability and Statistics
Jiheng Zhang
Due Apr 11, 2016
Solution to Practice Problem 8
1. (a) For independent variables X, Y ,
E[XY ] = E[X]E[Y ]
so
Cov(X, Y ) = E[XY ] E[X]E[Y ] = E[X]E[Y ] E[X]E[Y ] = 0
(b) As
The Hong Kong University of Science and Technology
probability
IELM 2510

Spring 2016
IELM 2510
Assigned May 1, 2015
Engineering Probability and Statistics
Jiheng Zhang
Due May 9, 2015
Practice Problem 12
1. (a) Sxx = 334.1, SY Y = 19.3, SxY = 66.9, SS = 5.9
(b) y = 0.2x + 5.7
(c)
H0 : = 0; H1 : 6= 0
r
(n 2)Sxx
B = 4.3
TS =
SS
t/2,n2 = t0.
The Hong Kong University of Science and Technology
probability
IELM 2510

Spring 2016
IELM 2510
Engineering Probability and Statistics
Assigned Mar 15, 2016
Jiheng Zhang
Due Mar 21, 2016
Practice Problem 6 Solution
1. Define Z =
X2
,
then Z N (0, 1).
(a)
Pcfw_X > k = Pcfw_Z > k 2 = 1 (k 2)
(b)
Pcfw_X 2 > 42 = Pcfw_X > 2 or X < 2
= Pcfw_Z
The Hong Kong University of Science and Technology
probability
IELM 2510

Spring 2016
M ODULE VII:
R EGRESSION
Jiheng Zhang
Spring, 2016
was run.
Introduction
Least Square Estimators
Statistical Inference
Transforming to Linearity
yi
i
xi
i
xi
1
100
45
6
150
2
110
52
7
160
3
120
54
8
170
4
130data63pairs9 (xi , 180
Example: Consider the fo
The Hong Kong University of Science and Technology
probability
IELM 2510

Spring 2016
M ODULE II B :
R ANDOM VARIABLES C ONTINUOUS
Jiheng Zhang
Spring, 2016
Introduction
PDF & CDF
Expectation
Variance
MGF
Comparison
Uniform
Exponential
Normal
Normal related
Mapping from Sample Space
D EFINITION
A random variable can be regarded as a mappin
The Hong Kong University of Science and Technology
probability
IELM 2510

Spring 2016
M ODULE VI:
H YPOTHESIS T ESTING
Jiheng Zhang
Spring, 2016
Introduction
Test in a Normal Population
Testing the Equality of Means
Test in Bernoulli Population
Test in Poisson Population
Introduction
In the previous module, we want to estimate the value of
The Hong Kong University of Science and Technology
probability
IELM 2510

Spring 2016
M ODULE V:
PARAMETER E STIMATION
Jiheng Zhang
Spring, 2016
Introduction
Point Estimate
Evaluating Point Estimate
Interval Estimate Normal
Interval Estimate Bernoulli
Introduction
In the previous modules, we know information about the distribution of the
p
The Hong Kong University of Science and Technology
probability
IELM 2510

Spring 2016
M ODULE III:
L IMITING T HEOREMS
Jiheng Zhang
Spring, 2016
Introduction
The Sample Mean and Sample Variance
Law of Large Numbers
The Central Limit Theorem
More Examples
Introduction
We need to study a large population!
Example:
The starting salary of HKUS