Summary of Counting Methods
The multiplication rule can be stated as, If there are n1 outcomes in experiment
1, n2 outcomes in experiment 2, and so on up to nk outcomes for experiment k,
then there are n1 x n2 x x nk outcomes if the k experiments are done

Lecture 10
Binomial Process
A series of trials in which each trial can have only one of two possible outcomes, such as
tossing a coin, is called a binomial process. A general formulation could be any
situation that can be answered with either a yes

Lecture 9 Random Samples, Statistics, and Central Limit Theorem
Independent random variables X1, X2, , Xn with the same distribution are called a random
sample. We use statistics to describe a sample. Examples of statistics include sample mean,
sample sta

Lecture 10 Normal Distribution and Lognormal Distribution
Normal Distribution
Most observations in real life follow the normal distribution. The reason can be
attributed to the central limit theorem which states, the sum of sufficient large n

Recitation 7
Binomial Process
If each trial is independent and the probability of a success on each trial is constant (p),
the number of successes in n trials, XB follows the binomial distribution, and it is called
a binomial random variable. It is

Residual Analysis:
1. Plot ei vs !
Check - Homoscedastic
- No trending
2. Plot ei vs ! in the model
Check - Homoscedastic
- No trending
3. Plot ei vs ei+1
Check - Homoscedastic
- No trending
4. Plot ei vs t
Check - Homoscedastic
- No trending
5. Plot Hist

Lecture 7 More than one variable
Joint Distributions
Last week we talked about discrete and continuous random variables, probability distributions,
expected values, and variances. Now we will look at problems that have

Lecture 12
Multinomial Distribution
The multinomial distribution is a more general case of the binomial distribution. In the
binomial distribution, you have only 2 possible outcomes from each trial. In the
multinomial distribution you have more than

Confidence Intervals and Hypothesis Testing of a Population Mean
(Variance Known)
Confidence Intervals
One-sided confidence level for lower bound,
=
One sided confidence interval for upper bound,
= +
Similarly, the two sided (1-) confidence interval f

Lecture 8 Functions of Random Variables
In many cases, a random variable is defined as the functions of other random variables. It will
be useful to know the properties of a random variable in terms of its constituent random
variable.
Y=X+c
Consider the s

E 243 Regression Recitation Problems
April 24, 2015
1) The regression equation of a line is = 0.444 + 0.798 X.
Coefficient
-0.444
0.798
Constant
X
Source of
Variance
Regression
Residual
Total
Degrees of
Freedom
t-stat
-0.76
9.97
Sum of Squares
55.268
Mean

1) Using the Spring 2015 survey data posted on Canvas, please answer the
following questions
1. Are there any outliers in Female Heights? (answer using Excel analysis)
2. Draw a histogram of Female Heights (in Excel)
3. Draw a pie chart showing the relati

Lecture 15
Hypothesis Test
An exhaustive list of possible hypotheses you can consider in the general form are,
H0
=
H1 -
0
H0
0
H1 -
>
0
H0
0
H1 -
<
0
0
Or
Or
In all these cases, you are checking whether your population mean is equal to or less
than, o

E243 Quiz 1 Spring 2015
You are allowed one help sheet (formulae, NO examples) and a calculator.
6+9 Points
1. How many automobile license plates can be made,
a. If each plate contains 2 letters followed by 4 digits.
b. Solve the same problem, if the two

Summary of Lectures 7 & 8
Joint Distributions
Continuous Variables
If the variables are continuous then the probability density function is given by,
( < < + < < + ) = (, )
Here f(x,y) is the joint probability density function of X and Y.
The probability

Summary of Lectures 5 and 6
Random Variables
The random variable is usually represented by an upper case letter, say X. A measured
value of the random variable is denoted by the corresponding lower case letter; in this
case x.
There are two basic kinds of

E 243 Quiz 2
Spring 2015
You are allowed 1 help sheet, statistics tables, and calculator.
(10+10+10 points)
1) Calls arriving to a freshman students cell phone can be modeled as a Poisson
process with 1 call per hour.
a) A freshman received 3 calls in the

E 243 Recitation 3 Spring 2015
February 13, 2015
Examples
1) Please answer the following questions
a. Are there any outliers in Male Heights? (answer using Excel analysis)
b. Draw a histogram of Male Heights (in Excel)
c. Draw a pie chart showing the rela

S
{5. 3) (53.1“) [51 F}
as: ass are
(H S} (H P) (H F)
am 0.14 are
(E S) (E P) (H F)
ass [:21 1119
It
CURE 1.25
:ibabilit}I values fer assemny line nperalitms
2.11 A faetery has twe assembly lines, eaeh ef whieh is shut
dawn (S), at partial caper:in (P), a

E 243 Regression Similar to HW5
Tuesday, April 21, 2015
2:56 PM
New Section 1 Page 1
New Section 1 Page 2
New Section 1 Page 3
New Section 1 Page 4
New Section 1 Page 5

Joint Distributions
Continuous Variables
f(x,y) is the joint probability density function of X and Y.
The probability that X lies in the interval (a,b) and Y lies in the interval (c,d) is obtained by
integrating the probability density function in these i

Recitation 3
Summary of Lecture
Random Variables
The random variable is usually represented by an upper case letter, say X. A measured
value of the random variable is denoted by the corresponding lower case letter; in this
case x.
There are two basic kind

E 243 Recitation 1 Fall 2016
September 2, 2016
Summary of Lectures 1 and 2
Important terms: Probability, Experiment, Event, Complement, Probability Axioms, Unions and
Intersections and the relation between them, Mutually Exclusive Events, and Exhaustive E