ENGG 2450A Probability and Statistics for
Engineers
SHAO, Shengcheng
27th Jan. 2016
SHAO, Shengcheng (CUHK)
ENGG 2450A Probability and Statistics for Engineers
27th Jan. 2016
1 / 14
Permutations
Choose r objects from n available distinct objects. Differen

Conditional Probability
If A and B are any events in S and P(B) 0,
the conditional probability of A given B is
P(A | B)
P(A B)
P(B)
Example :
Prob that a communication system has high fidelity is 0.81
Prob that it has high fidelity and high selectivity i

Chapter 3
Probability
1
Sample Space & Events
Sample space: in statistics, all the possible outcomes of
an experiment is called a sample space. In other words,
it encodes all possible outcomes of an experiment
An event is a subset of the sample space
A

Chapter 4
Probability Distributions
1
Random Variable
Definition any function that assigns a
numerical value to each possible outcome
Notations: X, Y and the possible values
they take on: x, y
Classification
Discrete RV take on finite / countably
infi

THE CHINESE UNIVERSITY OF HONG KONG
Department of Systems Engineering & Engineering Management
2015/2016
2nd Term
Course Code & Title:
ENGG2450C (ESTR2005) Probability and Statistics for Engineers
Instructor and Teaching Assistants
Instructor
Name
Office

Chapter 5
Probability Densities and Joint
Distributions
1
Joint Distributions
Experiments may be conducted where two random
variables are observed simultaneously in order to
determine their individual behavior, as well as the
relationship between them
P

Chi-Kong Ng, ENGG2450C/ESTR2005, Dept. of SEEM, CUHK
4:1
Chapter 4.
Discrete Random Variables and Probability
Distributions
Chi-Kong Ng, ENGG2450C/ESTR2005, Dept. of SEEM, CUHK
4:2
4.1. Random Variables
It frequently occurs that in performing an experime

Homework 1 Solution
1.2
Population: the collection of students preferences for rappers. (Here by population,
we mean statistical population)
Sample: the rappers named by the listener who call into the station.
The sample is not representative, since the a

ENGG2450A
Assignment 10 Solution
April 20, 2016
7.42
(a)
H0 : = 183
H1 : > 183
(b) Type II error. Because the true mean is 190, H0 is wrong.
If we reject H0 , we are right. If we accept H0 , we will make a type II error.
7.49
X100
X100
< z0.01 = 2.325.
<

ENGG2450A
Assignment 9 Solution
April 7, 2016
7.1
We can use t-statistic.
n = 30, s = 819.35, = 0.05, t/2 = t0.025 = 2.045 with d.f.=29.
The maximum error is
E = t0.025 sn = 2.045 819.35
= 305.9.
30
7.9
Since the sample size n = 80 is large, we can use z-

Solution to homework 4
4.20, 4.23, 4.46, 4.47, 4.56, 4.63, 4.64
4.20
The 15 components to be checked are independent and identical trials, the number
of fails follow Binominal Distribution.
(a) n = 15, p = 95%
P = 1 b(0; 15,95%) = 1 0.9515 = 0.537.
(b) n

THE CHINESE UNIVERSITY OF HONG KONG
ENGG2450A/B/C Probability and Statistics for Engineers
Mid-Term Examination
Instructions:
This paper must be answered in English. The duration of this paper is 1.5 hours.
Attempt ALL questions. The full marks of this

Chi-Kong Ng, ENGG2450C/ESTR2005, Dept. of SEEM, CUHK
Chapter 5.
Continuous Random Variables and
Probability Distributions
5:1
Chi-Kong Ng, ENGG2450C/ESTR2005, Dept. of SEEM, CUHK
5:2
5.1. Continuous Random Variables
Example:
Let X be a random variable wh

Chi-Kong Ng, ENGG2450C/ESTR2005, Dept. of SEEM, CUHK
3:1
Exercise:
Electrical engineers are considering two alternative systems for
sending messages. A message consists of a word that is either a 0
or a 1. However, because of random noise in the channel,

1
Chapter 1.
Introduction
2
Engineering Method
An engineer is someone who solves problems of interest to society
by the ecient application of scientic principles.
Engineers accomplish this by either rening an existing product
or process or by designing a

ENGG 2450A Probability and Statistics for
Engineers
SHAO, Shengcheng
20th Jan. 2016
SHAO, Shengcheng (CUHK)
ENGG 2450A Probability and Statistics for Engineers
20th Jan. 2016
1 / 12
Statistics
Uncertainty versus Destiny
Make use of data we do have
Predict

ENGG2450A Tutorial 5:
Continuous random variables &
Probability densities
XU Tianrun
trxu@se.cuhk.edu.hk
Continuous Random Variables
A random variable X is continuous if its
distribution function = P(X ) =
for some integrable [0, )
Definition: The fun

ENGG 2450A TUTORIAL 3
- BAYES THEOREM
Terminologies Review
Joint Probability: the joint probability of event A and event B
occurring concurrently is denoted as P(A B) or P(AB).
Conditional Probability: P is called the conditional probability
of event A

Tutorial 4
Probability distribution
ENGG2450A zhouxiang
zhouxiang203@gmail.com
Geometric Distribution consider the number of the trials on which the first success
occurs.
; = (1 )1 , = 1,2,3
=
=1
=
(1 )1 = +
=
=1
so
1 =
=1
=1 (1
1 =+
1
=
1 ( + 1)

ENGG2450A Tutorial 7:
Description of Data & Sampling
XU Tianrun
trxu@se.cuhk.edu.hk
Outlier
No rigorous definition
May have huge influence on our data analysis
How can we find the outliers?
Intuitive methods (plots,etc.)
Mathematical methods (Cooks distan

Chapter 6
Random sample (sample data) drawn from a population
Population may be finite or infinite
Certain quantities determined from samples can be
expected to vary from sample to sample
Sampling distribution distribution of a statistic
calculated on

Chapter 4: Probability Distributions
Statistical problems concerned with one or
a few numbers associated with outcomes
of experiments
Number of defectives in inspection of
manufactured product
Lifelength of a battery
1
Random Variable
Definition any f

Geometric Distribution
Series of Bernoulli trials
Consider number of trials until first success
g(x;p) p( 1 p)
x 1
x 1,2,3.
1
Mean
p
1
Negative Binomial Distribution
Total # Bernoulli trials, X, to obtain r successes
(r-1) successes occur in the firs

Chapter 3
Probability
1
Probability
Questions of interest:
What do we mean when we say that the probability of an event is
0.5, 0.02, 0.81?
How do we determine and measure the probabilistic values?
What are the mathematical rules that the probabilitie

Chapter 2
Organization and Description of data
1
Treatment of Data
Statistical data collected from surveys, experiments,
etc. are numerous
Description and Analysis: need descriptive picture for
effective representation upon which we can perform
analysi

ENGG2450A Probability and Statistics for Engineers
Spring 2016
Prof. Xuefeng Gao
SEEM
1
Course Administration
Course Outline
Course website on https:/elearn.cuhk.edu.hk
Lecture notes by courtesy of Prof. Helen Meng
Textbook (required):
R. Johnson, I.

THE CHINESE UNIVERSITY OF HONG KONG
Department of Systems Engineering & Engineering Management
2015/2016
2nd Term
Course Code & Title:
ENGG2450A Probability and Statistics for Engineers
Instructor and Teaching Assistants
Instructor
Name
Office
Telephone
E

Homework 7 Solution
2.14
2.15
2.54
(a) We can estimate the mean and median in this case since we know the class
marks and frequencies for each class.
(b) We cannot estimate the mean in this case, since we dont know the class
marks for less than 90 and mor