Statistics 321 Lab #5: Continuous Random Variables. Uniform and Normal
Distributions.
Working with Probability Density functions (f (x)s) and Cumulative Distribution Functions (F (x)s)
Lab Exercise 1. A random variable X has the probability density functi

Statistics 321 (L01): Notes and Class Examples of May 15
Prior to introducing the notion of probability, there are some terms that need to be defined. These are:
1. random experiment
2. sample space, or (S)
3. event.
Definition: A random experiment is a p

Statistics 321 Lab #4: Solutions
Solutions:
1. (a) Defining a random variable X to count how many, out of n = 40, disagree with internet companies
handing over private information to authorities. Each person is treated as a Bernoulli trial (disagree is a

Linear Functions of Random Variables
Let ! , ! , , ! represent random variables where ! = ! and ! = ! < ,
for = 1,2, , . Note that these variables may but not have to be independent of each
other.
Then for ! and ! , where (dont compare a variable to itsel

Introduction to Probability
In everyday conversation, the term probability is a measure of ones belief in the
occurrence of a future event. We accept this as a meaningful and practical interpretation
of probability but seek a clearer understanding of its

Probability Proofs
If A is any event dened on a sample space S, the symbol P(A) will denote the
probability of A, and we will refer to P as the probability function.
It is, in effect, a mapping from a set (i.

Stat 321 Quiz 2 L02 (Robison, Fall 2014) October 9
Name: K51 (Please, also include your name on the last page as well!)
Student ID: ME"
This is a closed book exam! Only Scientific calculators are permitted [no prog/ graph)
Show all relevant work [formulae

l:'_ " UNIVERSITY OF
JCALGARY
UNIVERSITY OF CALGARY
FACULTY OF SCIENCE
DEPARTMENT OF MATHEMATICS AND STATISTICS
MIDTERM EXAMINATION
Stat, 321 (Robison) - Fall, 2014
DATE: 23/0ctober/2014 Time: 50 Minutes
Student ID Number:
EXAMINATION RULES
T otal Actual

UNIVERSITY OF CALGARY
FACULTY OF SCIENCE
DEPARTMENT OF MATHEMATICS AND STATISTICS
MIDTERM EXAMINATION
Stat, 321 (Robison) Fall, 2014
DATE: 23/October/2014
Time: 50 Minutes
Student ID Number:
Last Name:
First Name

!1
Extra Practice questions for quiz 2
1. Suppose the values below represent the grade point averages, corresponding gender
and satisfaction with their program (scale of 1 to 5 with 1 representing unsatisfied and
5 representing very satisfied) of 26 rando

Stat 321 Practice Quiz 1 L02 (Robison, Fall 2015)
Due Sunday Sept 27 11:59pm
Please submit your answers via TopHat for bonus marks. Understand these questions may/may not
be similar to that of the actual quiz on October 1. Ple

Stat 321 Quiz 2 L02 (Robison, Fall 2014)
October 9
Name:_ (Please, also include your name on the last page as well!)
Student ID:_
This is a closed book exam! Only Scient

Statistics 321 Lab #5: Working with Continuous Random Variables; the
Normal Distribution and R
Questions 1 is a continuation of Question 4 from Lab #4. Complete Questions 1, 2, 3 and 4 in the lab.
1. In Lab #4, Question 4 you were exposed to the following

4.6
4.8
a.F(i)=P(Y50= l -P(Y>i)= l -P(l"itrialsarefailures)= l -q.
b. It is easily shown that all three properties hold.
a. The constant k = 6 is required so the density function integrates to l.
b.P(.4s Y5 l)=.648.
c. Same as part b. above.
d. P(Ys .4 |

5.79 Referring to Ex. 5.16, integrating the joint density over the two regions of integration:
0 l+y1 l l-yi
E(Y.Y2)=I Iylyzdyzdyi Iylyzdyzdyl =0
-I 0 0 0
5.80 From Ex. 5.36, fl(y,)= yl +% , 05y] S 1, and f2(yz)= y2 +%, 05y2 S 1. Thus,
E(Y1)= 7/12 and E(Y

3.36
3.37
3.38
3.40
3.41
a. The random variable Y does not have a binomial distribution. The days are not
independent.
b. This is not a binomial experiment. The number of trials is not xed.
a. Not a binomial random variable.
b. Not a binomial random varia

L413 .
Statistics 321: Discrete Distributions and R
To download R for home or laptap use, visit http:/www.rproject.org, on the lefthand side under the
heading Download click 011 CRAN. Select either of the Canadian links. Under the heading Download and
Ins

Solutions for Lab #2 Stat 321
1
Key formulas to know:
n
finite Geometric series : For | a |< 1, a i =
i=0
a n+1 1
a 1
infinite Geometric series : For | a |< 1, a i =
i=0
1
1 a
1. (a) Defining S to represent the event that a person (aged 25 or older) is a

Solutions to an example from the bonus assignment
Example
A late-night stargazer sees an object, that is unidentifiable, crash. This person is sure the object
landed in one of three areas; namely: area 50, area 51, or area 52.
Unfortunately, the person is

Some Key Formulas for the Midterm
Law of complement: P(A c ) = 1 P(A)
addition law: P(A B) = P(A) + P(B) P(A B)
Total Law of probability: P(A) =
n
n
P(A | B )P(B ) = P(A B )
i
i=1
i
i
i=1
Total Law of probability simplified: P(A) = P(A B) + P(A B c )
Dem

Some Key Formulas for Quiz 2
Law of complement: P(A c ) = 1 P(A)
addition law: P(A B) = P(A) + P(B) P(A B)
Total Law of probability: P(A) =
n
n
P(A | B )P(B ) = P(A B )
i
i
i
i=1
i=1
Total Law of probability simplified: P(A) = P(A B) + P(A B c )
Demorgan

Statistics 321 (L01): Independent Events; Conditional Probability (May 22)
Independent Events
Definition: Two events, A and B are said to be independent events if and only if the occurrence of one
event (A) does not effect the occurrence of event B. If tw

Statistics 321 (Spring 2015) : Practice Problems Covering Set Operations and
Introduction to Probability
Please attempt the following questions 1, 2, and 3 in the lab of May 14th. The answers are found on the last
page of the lab worksheet. If you have qu

Statistics 321 (Spring 2015) : Linear Functions of Random Variables
Result: Let X1 , X2 , , Xn represent random variables where E(Xi ) = i and V ar(Xi ) = i2 < ,for
i = 1, 2, , n. Additionally, Xi and Xj , for i 6= j, are not necessarily independent rando

Statistics 321 (Spring 2015): Continuous Random Variables
Definition: Let X be a continuous random variable. A function f (x), defined over the set of all real
numbers, is called a probability density function of X if and only if
1. f (x) 0 for any x.
2.

Statistics 321 (Spring 2015): Moment Generation Functions
Definition: The ith-moment of a random variable X taken about its origin is defined as:
X
xi P (X = x)
E(X i ) =
all x
For example, the
1. the 1st moment of X is E(X 1 ) = E(X) = X .
2. the 2nd mom

Statistics 321 (L01) - Counting Methods
Multiplication Principle: Consider the following events, A1 , A2 , , Ak , where A1 can occur in a-ways,
A2 can occur in b-ways, , Ak can occur in k-ways. The number of ways in which A1 , A2 , , Ak can occur
together