Probability Theory - Advanced Engineering Math - Tutorial Slides, Slides of Engineering Mathematics

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Topics of Probability Theory

• If Pr(B) is nonzero:
• Brief review

The Bayes’ rule：

• The mean or expected value of a random variable

with pmf f (xj):

• E(X)=
• Brief review
• Mean (Expectation) of a Distribution

• For a discrete random variable X with mean μ, the

variance of X is defined as the expected deviation from the mean:

• Var(X)=
• Brief review
• Variance of a Distribution

• Var(c X) = c^2 Var(X)
• Var(X+Y) = Var(X) + Var(Y)
• In the second property, X and Y must be independent.
• Give a counterexample?
• Brief review
• Properties of Mean and Variance

• If a random variable X has mean and variance then

the random variable

• has the mean and variance as (p.1037)
• Brief review
• Transformation of Mean and Variance

Question 1 ( Q3 in Final 2010-2011 ) :

Solution:

or

Question 2 ( Q5 in Final 2010-2011 ) :

If two-thirds of the books in the library are new, and half of the books in

the library are non-fiction, but only one-third of the new books are non-

fiction, find the probability that a book selected randomly from the

library is new, given that it is a non-fiction book.

Solution: Suppose that the event the book is new is A, the event the

book is non-fiction is B. Then

P(A)=2/3; P(B)=1/2; P(B|A)=1/

From Bayes’ rule：

P(A|B)=P(B|A)* P(A)/P(B)=1/3*2/3 / ½ = 4/

Question 3 ( Q7 in Final 2010-2011 ) :

Solution:

a) no sections receive a hard test: 224 / 304

b) exactly one section receives a hard test: 81 223 / 304

c) the expectation of the number of sections that receive a hard

test:

8 1

22 3 30 4

8 2

22 2 30 4

8 3

22 1 30 4

8 4 30 4

Question 4 ( Q4 in Final 2011-2012 ) :

The populations of Hong Kong, Shanghai, and Beijing are 7

million, 13 million, and 10 million, respectively.

(a) Suppose we know that Alice is from one of these three cities.

What is the probability that Alice is from Hong Kong?

(b) Eight out of ten Hong Kong residents prefer Coke to Pepsi, two

out of three Shanghai residents prefer Pepsi to Coke, while Beijing

residents are evenly split in their preferences (suppose that each

resident in each city has a preference). We also learn that Alice

prefers Coke to Pepsi. Now what is the probability that Alice is

from Hong Kong?