Relevant videos for practicing for the Mathematics entrance test

for the LLM Master in Law and Finance programme

- https://www.youtube.com/playlist?list=PLM-mb7IpX4moTbG2Pe96z2JcQcmIeFYIr
A playlist with mathematics test training videos

- https://www.youtube.com/watch?v=mk8tOD0t8M0
A very simple video, using numerical examples to explain Mode, Median, Mean, Range, and
Standard Deviation

- https://www.youtube.com/watch?v=qqOyy_NjflU
A simple video showing you how to calculate Standard Deviation and Variance, providing a
numerical example with 6 observations. (This video uses a sample variance formula)

- https://www.youtube.com/watch?v=sOb9b_AtwDg
This video can help you distinguish sample and population variance. More theories are also
explained in this video, telling you what variance actually is

- https://www.khanacademy.org/math/probability/probability-geometry#probability-basics
This video shows the basics of probabilities

- https://www.youtube.com/watch?v=OvTEhNL96v0
This video shows information about Expected Value and Variance of Discrete Random
Variables with numerical examples

- https://www.khanacademy.org/math/ap-statistics/random-variables-ap/discrete-random-
variables/v/variance-and-standard-deviation-of-a-discrete-random-variable
This video shows a numerical example of calculating variance and a standard deviation of a
discrete random variable (5 variables)

Below you will find “Introduction to statistics and probability theory”

slides for math test preparation purposes

Also note the Trial Exam on our website which is of course

strongly recommended in order to practice for the math test
Introduction to statistics and probability theory
This set of notes is based on Chapters 1,4, 6, 7 of Keller, Gerald, Statistics for
Management and Economics, 9th edition. ISBN: 978-1111527327.
SOME BASICS
 Some Basics
2

 1. Some Basics
 1.1 Statistics vs. Statistic
 Statistics as a field of mathematics dealing with the
collection, explanation and interpretation of data.
 A statistic is a measure that tries to capture some
information about the data set.
 Examples: mean, median, standard deviation, correlation,
covariance, sample estimate of the population mean, …
 (The plural form of statistic is also called “statistics”!)
 Some Basics
3

 1. Some Basics
 1.2 Population vs. Sample
 The population of a study is the group
of all items of interest in that study.
 A sample is a subset of the population that
is studied. (A desired characteristic is that its
data was obtained randomly.) “Sampling
Process”
 The “sampling process” is the method by
which we select observations from the
population to arrive at a sample.
 Typically some form of random sampling.
 Some Basics
4

 1. Some Basics
 1.3 Parameters vs. Statistics
 Descriptive measures about the
population (e.g., the population mean)
are called parameters.
 Descriptive measures about a sample are
called statistics. “Sampling
Process”
 We typically use latin-based letters for
sample statistics.
 Example: the (sample) mean
(pronounced “x-bar”)
 Some Basics
5

 1. Some Basics
 1.4 Data Types
We differentiate between the following data types:
 1. INTERVAL or QUANTITATIVE data
 Real numbers
 Distances between values with intrinsic meaning.
 Examples:
 On previous slide: Time in seconds

(distance meaningful: 60 seconds twice as long as 30 seconds)

 Height of students

 Length of cars

 …
 Some Basics
6

 1. Some Basics
 1.4 Data Types
We differentiate between the following data types:
 1. INTERVAL or QUANTITATIVE data
 2. ORDINAL data
 An ordered ranking among data exists.
 Distances do no have intrinsic meaning.
 Ex. on previous slide : Song rating: {bad, average, good, excellent}
 We can assign numbers to each value, but we need to maintain
the order, e.g., bad=1, average=2, good=3, excellent=4.
 Also possible : bad=0, average=23; good=34;
excellent=100; distances between values do not matter.
 Some Basics
7

 1. Some Basics
 1.4 Data Types
We differentiate between the following data types:
 1. INTERVAL or QUANTITATIVE data
 2. ORDINAL data
 3. NOMINAL or CATEGORICAL data
 Values have no order, nor any intrinsic numerical value;
any number can be applied to represent a value.
 Examples:
 On previous slide: Artist, Album, Genre

 Family status = {single, married, divorced, widowed}

 Gender = {male, female}  No difference b/w {male=0,

female=1} or having {male=1, female=0}
 Some Basics
8

 1. Some Basics
 1.4 Data Types
In fact there is a hierarchy among data types:
 Example: Exam scores
Interval data
 Exam scores (interval data) is often compressed into letter
grades (ordinal data): 94 points = A
 Letter grades (ordinal data) can be further compressed into
Ordinal data simple pass/fail categories (nominal data): A = Passed.

 Note: Moving from a higher level to the lower level, we lose

Nominal data
information, which we cannot undo, so going back from a lower
level to a higher level is not any more possible:
 Student had a B  How many points did she have?

 Student passed  Which letter grade/score did he have?

 1

DESCRIPTIVE STATISTICS
 Descriptive Statistics
2

 1. What is Descriptive Statistics?

 Question:
 Suppose you have some data set and you would like to
tell someone about it without having to show him each
single data point. What do you do?
 (Or, the data set is simply so large that you get a
headache from looking at thousands of values.)

 Answer:
 You may try to come up with some
summary statistics that somehow Simply put, all that is
describe the data set. descriptive statistics.
 Descriptive Statistics
3

 1. What is Descriptive Statistics?

 Examples:
 Measures of Central Location
 Mean (arithmetic, geometric), Mode, Median
 Measures of Dispersion (or Spread, Variability)
 Range, Interquartile Range, Mean Absolute Deviation (MAD)
 Standard Deviation, Variance
 Coefficient of Variation
These are the most
 Measures of Relative Standing commonly used ones.
 Percentiles (Quartiles, Quintiles, Deciles) There is nothing like a “right”
vs. “wrong” measure. You can
 z-Score also come up with your own
new measures if you like to
 Measures of Linear Relationships that seem better to you!

 Covariance, Correlation
 Descriptive Statistics
4

 1. What is Descriptive Statistics?

 Population versus Sample
 The population of a study is the group
of all items of interest in that study.
 A sample is a subset of the population
that is studied.
 Note that the mean of the sample of course “Sampling
differs from the mean of the population – in Process”
(next chapter)
fact, every time we take a sample, its mean
will slightly be different.
 We need to differentiate between the
sample mean and population mean.
 Descriptive Statistics
5

 1. What is Descriptive Statistics?

 Take-away point thus far:
 Since population parameters are different from sample
statistics, there are (sometimes) different formulas for
computing population versus sample statistics!
 Descriptive Statistics
6

 1. Let’s work with an example:

 Suppose we have the following data set: a random
sample of 15 final scores of last semester:
 859,1018, 422, 813, 823, 912, 824, 643, 1013, 874,
929, 912, 655, 778, 629

 Q1: What sort of measures would you want about

those scores to describe this data set?

 Q2: What sort of charts would be helpful?

 Descriptive Statistics
7

 2. Measures of Central Location

 (A) Arithmetic Mean
An aside:
Σ (which is the Greek capital letter of
Sample: “Sigma”) is the summation sign used in
math to sum up several terms. “i" is the
index variable running from 1 to “n” (n
would be in our case 14).

Population:
We will see the Sigma summation sign
Little “n” is used for sample sizes and over and over again; make sure you
capital “N” for the population sizes. understand it!
 Descriptive Statistics
8

 2. Measures of Central Location

 (A) Arithmetic Mean
Our Sample: 859, 1018, 422, 813, 823, 912, 824, 643,
1013, 874, 929, 912, 655, 778, 629.
 Descriptive Statistics
9

 2. Measures of Central Location

 (A) Geometric Mean
 Thearithmetic mean does not work in examples where we
deal with growth rates or, generally, some rate of change.
 Example:
 Suppose you invest $1,000 for 2 years.
 In 1st year it grows 100% to $2,000. (R1=100%)
 In 2nd year it suffers a 50% loss, back to $1,000. (R2=-50%)

Arithmetic mean = (R1 + R2)/2 = (100+(-50))/2 = 25%

But investment started with $1,000 is again at $1,000  0% growth!
 Descriptive Statistics
10

 2. Measures of Central Location

 (A) Geometric Mean

 Rg = ((1+100%)*(1-50%))^(1/2) -1 = 0

 InExcel, use 1+R for growth rates and then use

function “=GEOMEAN(cell area)-1” to compute
the average growth value.
 Descriptive Statistics
11

 2. Measures of Central Location

 (B) Median
The “middle value” in the ordered data set, i.e. the value
at which same number of data points above and below.
Step 1: Sort data from small to large values.
Step 2: Find middle value:
If odd number of data points, then median is at
position (n+1)/2. If even number, then take the
(arithmetic) mean of the two “middle” numbers.

The median only exists for interval or ordinal data

(because nominal data such as “color of the eyes”
has no intrinsic ordering and cannot be sorted).
 Descriptive Statistics
12

 2. Measures of Central Location

 (B) Median
Formally:

Sample:

Population:
where D(.) is the cumulative distribution function*

* That means that half of the values of the data set are below the value x. More on this in chapter 6.
 Descriptive Statistics
13

 2. Measures of Central Location

 (B) Median
In our sample:
Step 1: Sort the data {422, 629, 643, 655, 778, 813,
823, 824, 859, 874, 912, 912, 929, 1013, 1018}.
Step 2: Find middle value:
15 observations
 odd number of data points
 median is at position (n+1)/2 = 8
 8th value in data set = 824 = Median
 Descriptive Statistics
14

 3. Measures of Dispersion
 (B) MAD, Variance & Standard Deviation
 More commonly used are MAD, Variance and Standard
Deviation.

 To derive their formulas, lean back and think for a

minute how you would create a measure that somehow
represented the variation in a data set.
 Descriptive Statistics
15

 3. Measures of Dispersion
 (B) MAD, Variance & Standard Deviation
 One intuitive way could be to measure the spread of the
data as the average distance of the data points from the
center of the data set.
 In math terms that would be:
“m” is some measure
of center – could be
1. Numerator sums up all the the arithmetic mean
distances between each or the median.
observation the center.
2. The denominator divides by
the number of observations to
get the average distance.
 Descriptive Statistics
16

 3. Measures of Dispersion
 (B) MAD, Variance & Standard Deviation
 Suppose we use the arithmetic mean of the population (μ)
as the measure of center, so:
 Descriptive Statistics
17

 3. Measures of Dispersion
 (B) MAD, Variance & Standard Deviation
 Suppose we use the arithmetic mean of the population (μ)
as the measure of center, so:
Issue: The numerator by
definition is 0 if we sum up
the differences between each
value to the overall mean.
Problem: the distances to
values below the mean and to
values above the mean cancel
each other out!
 Ex: data set {10,20,30}.
 Descriptive Statistics
18

 3. Measures of Dispersion
 (B) MAD, Variance & Standard Deviation
2 solutions to get rid of the problem of distances
canceling one another out:
 1. We add up the absolute values of the distances.
= Mean absolute deviation (MAD)*
of the population

 2. We add up the squared distances:

=Population Variance

* There is a variationsof this formula (the median absolute deviation), where the median is used instead of μ.
 Descriptive Statistics
19

 3. Measures of Dispersion We cannot interpret

squared dollars
 (B) MAD, Variance & Standard Deviation easily, so let’s take
the square root to
 The mean absolute distance (MAD) here: get back the
variation in dollars.
6.67 units (e.g., dollar per hour salary)
 The variance here: 66.67 units (squared dollars!!)
 Descriptive Statistics
20

 3. Measures of Dispersion
 (B) MAD, Variance & Standard Deviation
 The square root of the population variance is
called “standard deviation” (called, “sigma” σ):
= Population standard deviation.
(we have “readable” units again).
 Descriptive Statistics
21

 3. Measures of Dispersion
 (B) MAD, Variance & Standard Deviation
 Recall that we said earlier that sample statistics are
different from population statistics – this is the case for
the variance/standard deviation:
Sample Population

Variance

Standard
Deviation
 Descriptive Statistics
22

 3. Measures of Dispersion
 (B) MAD, Variance & Standard Deviation
 How to compute the MAD, Variance and Std Deviation?
1. By hand (if data set not too large), or
2. Variance and Std Deviation using Excel

Excel formula for MAD is “AVEDEV(range)”

3. With your calculator (differs by model).

 Descriptive Statistics
23


 Descriptive Statistics
24

 3. Measures of Dispersion
 (C) Coefficient of Variation
 Is a std dev of 161 points among final scores large? The
standard deviation by itself cannot be interpreted when
the magnitude of the variable under question is unknown.

 The Coefficient of Variation simply scales the standard

deviation by the mean:

Population Sample

 d
 Descriptive Statistics
25

 3. Measures of Dispersion
 (C) Coefficient of Variation
 For our sample of final scores of last year:
The variation among the
final scores is about 20%
of the mean score. This is
not a large variation.

 (+) While std dev across different data sets cannot be

compared, the coefficient of variation can be.
 (-) If the mean is close to 0, then CV explodes, rendering
the coefficient of variation meaningless.
 D
 1

PROBABILITY
 Probability
2

 1.1 Basic Terminology: Random Experiment

A random experiment is any process that leads to
one or several basic outcomes (or a list thereof).*
 Examples:
 1. Flipping a coin An EXPERIMENT is called
“random” if its RESULT cannot
 2. PA Lottery “Treasure Hunt”
be predicted with certainty.
 3. Playing football/baseball
 4. Predicting short-time stock price movements

* Basic outcomes need to be mutually exclusive. (If we consider the weather as a random experiment,
then for example “sunny” and “rainy” could not be basic outcomes as both can occur simultaneously.)
 Probability
3

 1.1 Basic Terminology: Random Variable

A random variable (RV) is a variable that takes on
the value of the outcome of an experiment.
 (We usually use X, Y, Z as the letter to designate a RV.)
 Examples:
A VARIABLE is called
 1. Let X be the RV for flipping “random” if its VALUE cannot
1 coin  X can take on the be predicted with certainty.

value “Heads” or “Tails”

 2. Let Y be the RV for the PA Lottery “Treasury Hunt”
 Y is a set of 5 numbers, each of which between 1 to 30.
 Probability
4

 1.1 Basic Terminology: Sample Space

 The sample space of a random experiment is
the list of all possible basic outcomes:

 Examples:
countable

 1. Flipping coins { Heads, Tails }

 2. PA Lottery “Treasure Hunt” { (1,2,3,4,5), …, (26,27,28,29,30) }
uncountable

 3. Football scores { …, (24:14), … }

 4. Batting averages { …, 0.25, …, 0.39, … }
 Probability
5

 1.1 Basic Terminology: Probability

A probability is the chance of one (or several)
outcome(s) of the sample space occurring.

A valid probability needs to satisfy 2 requirements:

 1. Needs to lie between 0% and 100%:
 2. The probabilities of all outcomes in a sample space need
to add up to exactly 1 (=100%).
 Probability
6

 1.1 Basic Terminology: Events

 An event is a set of one or more outcomes from the
sample space. The probability of an event is the sum
of probabilities of all outcomes of the set.

 Example:
 Experiment: Rolling a die once
 Sample Space: {1,2,3,4,5,6}
 Probability of any number (by classical approach): 1/6.
 Event: Rolling an odd number with a die = {1, 3, 5}
 Pr (odd number) = 1/6 + 1/6 + 1/6 = ½ = 50%.
 Probability
7

 1.2.1 Venn Diagrams

A Venn diagram is a fancy name for a box
that represents the sample space. John Venn
(Brit, 1834-1923)

 Helps in exercises with probability rules/operators.

Size normalized to 1 (100%)

 Probability
8

 1.2.1 Venn Diagrams

 Regions
within the box represent events with their
area equal to the probabilities of those events.
 We typically use circles for generic events (but we
don’t have to, see next slide)
 The area of the event represents its probability.

A B
How does the Venn
diagram look like
for the random
experiment “Rolling
a die once”?
 Probability
9

 1.2.1 Venn Diagrams

 Example: “Rolling a die once”
 Each event has the same probability to occur
(classical approach), so the areas are of same size.

Rolling a 1 Rolling a 2: Rolling a 3

Pr (“1”) =
Pr (“2”) = 16.7% Pr (“3”) = 16.7%
1/6=16.7%

Rolling a 4 Rolling a 5 Rolling a 6

Pr (“4”)=16.7% Pr (“5”) = 16.7% Pr (“6”) = 16.7%
 Probability
10

 1.2.1 Venn Diagrams

 The“outside” of an event is called the complement
of an event.

A
Complement of A,
written as A or AC and
called “A not” or
“A complement”
 Probability
11

 1.2.1 Venn Diagrams

 Example: Rolling a die
 Event: If A = “Rolling a 1”, then AC=“Not rolling a 1”

Rolling a 1
Pr (“1”) = 16.7%

Not rolling a 1
Pr(“not rolling a 1”) =
100%-16.7% = 83.3%
 Probability
12

 1.2.1 Venn Diagrams

 Theintersection of 2 events A and B is the event when
both events occur simultaneously (call it “C”).
 Note: A and B are not basic outcomes, so events can
occur simultaneously.
Say, A=rolling
an odd number;
B=rolling a
A B number > 3.
C
Pr (A)=1/2
Pr(B)=1/2
C=rolling a 5
Pr (C) =1/6

C:= Pr (A and B)
 Probability
13

 1.2.2 Contingency Tables

 Suppose we are interested if gender discrimination occurs
at a company and have the following 5-year personnel
data set of 200 workers with their gender & promotions.
1. Female; Promoted … 198. Male; Not promoted
2. Male; Promoted … 199. Male; Promoted
3. Female; Not promoted … 200. Female; Not promoted

 Step 1: We can create an absolute frequency table.

Promoted Not Promoted
Female 6 24
Male 34 136
 Probability
14

 1.2.2 Contingency Tables

 Ex: Gender discrimination at a company
 Step 2: We transfer the absolute frequency contingency
table into a relative frequency contingency table.
 (We just learned to do this in Excel in Chapter 3, using Pivot Tables.)

Promoted (B1) Not Promoted (B2)

Female (A1) 0.03 0.12
Male (A2) 0.17 0.68
 Probability
15

 1.2.2 Contingency Tables

 Ex: Gender discrimination at a company.

Marginal
Joint Probabilities
Probabilities
Promoted (B1) Not Promoted (B2)
Female (A1) 0.03 0.12 0.15
Male (A2) 0.17 0.68 0.85
0.20 0.80 1.00

Marginal Probabilities Has to add up to 1

= the sum of (=100%)
rows/columns
 Probability
16

 1.2.2 Joint and Marginal Probabilities

 Marginal Probabilities = Sum of Rows or Columns
 Pr (Female) = Pr (Female AND Promoted) +
Pr (Female AND Not Promoted)
= 0.03 + 0.12 = 0.15

Promoted (B1) Not Promoted (B2)

Female (A1) 0.03 0.12 0.15
Male (A2) 0.17 0.68 0.85
0.20 0.80 1.00
 Probability
17

 1.2.2 Joint and Marginal Probabilities

 Thejoint probabilities are obtained by multiplication of
marginal probabilities:*
 Pr (Promoted AND Female) = 0.20 * 0.15
= Pr (Promoted)*Pr (Female)
= 0.03
Promoted (B1) Not Promoted (B2)
Female (A1) 0.03 0.12 0.15
Male (A2) 0.17 0.68 0.85
0.20 0.80 1.00

* This only works if the events are independent – more on this below!
 Probability
18

 1.2.2 Joint and Marginal Probabilities

 How does the Venn diagram look like for our gender
discrimination example?
 1. Marginal Probabilities

20%
promoted

80%
Female Male not promoted
(15%) (85%)
 Probability
19

 1.2.2 Joint and Marginal Probabilities

 How does the Venn diagram look like for our gender
discrimination example?
 2. Joint Probabilities

Promoted AND
female = 3% Promoted AND male = 17%
20%
promoted
Not
promoted
AND Not promoted AND male = 68%
female = 80%
not promoted
12%
 Probability
20

 1.2.3 Conditional Probabilities

 To get back to our discrimination example:
Promoted Not Promoted
Female 0.03 0.12 0.15
Male 0.17 0.68 0.85
0.20 0.80 1.00

 To
answer our question whether there is discrimination:
Shall we simply compare joint probabilities?
 “17% versus 3% of all 20% promotions go to men. So men are
clearly favored.” ?
 Probability
21

 1.2.3 Conditional Probabilities

 Discrimination example:
Promoted Not Promoted
Female 0.03 0.12 0.15
Male 0.17 0.68 0.85
0.20 0.80 1.00

 No, because 85% of all workers are male, naturally we

would expect a higher share of promotions to go to men
given that there are so many more men in the company!
 Probability
22

 1.2.3 Conditional Probabilities

 Instead, we should compare: Conditional
probabilities
 What is the probability of being …
male
female
given that one is promoted?
 If there was no discrimination, we would expect the
that men are promoted 85% of the time (as they are
constitute 85% of the workers) and women 15%.
 In other words: if there was no discrimination we would
expect that “being promoted” is independent from So how do
we compute
gender “male/female”. conditional
probabilities?
 Probability
23

 1.2.3 Conditional Probabilities

 Using the Venn Diagram to find the conditional prob.:
Pr (B|A) = Pr (B GIVEN that A is true)
= Pr (area of B GIVEN that we are “inside” A)
= Pr (dark blue as a share of light blue area)
= Pr (A and B) / Pr (A)

A B
A and B
 Probability
24

 1.2.3 Conditional Probabilities

 Gender discrimination example:
 Let event A=Gender, event B=Promoted and simply plug in:

Promoted (B) Not Promoted (BC) 85% and 15%

Female (A) 0.03 0.12 0.15 coincides with the
share of male and
Male (AC) 0.17 0.68 0.85 female workers.
No evidence of
0.20 0.80 1.00 gender discrimi-
nation.
 Probability
25

 1.2.3 Conditional Probabilities

 What we just showed: the event “male/female” is
independent from the event “promoted/not promoted”!

Promoted (B) Not Promoted (BC)

Female (A) 0.03 0.12 0.15
Male (AC) 0.17 0.68 0.85
0.20 0.80 1.00
 Probability
26

 1.2.3 Conditional Probabilities

 Definition: Two events are said to be independent if

(or since the choice of the letters are arbitrary:

)

In our example:

 Probability
27

 1.2.3 Conditional Probabilities

 How would no independence look like in our example?
Promoted (B) Not Promoted (BC)
Female (A) 0.03 0.07 0.12 0.15
Male (AC) 0.17 0.68 0.85
Change just one
0.20 0.80 1.00 joint probability…
 Probability
28

 1.2.3 Conditional Probabilities

 How would no independence look like in our example?
Promoted (B) Not Promoted (BC)
Female (A) 0.03 0.07 0.12 0.08 0.15
Male (AC) 0.17 0.13 0.68 0.72 0.85
0.20 0.80 1.00

Then, to maintain the marginal

probabilities, all the other joint
probabilities need to change
accordingly.
 Probability
29

 1.2.3 Conditional Probabilities

 How would no independence look like in our example?
Promoted (B) Not Promoted (BC)
Female (A) 0.07 0.08 0.15
Male (AC) 0.13 0.72 0.85
0.20 0.80 1.00

Now women receive 35%of all

promotions even though they are
just 15% of the workers.
 No independence any longer!
 Probability
30

 1.2.3 Conditional Probabilities

 How would no independence look like in our example?
Promoted (B) Not Promoted (BC)
Female (A) 0.07 0.08 0.15
Male (AC) 0.13 0.72 0.85
0.20 0.80 1.00
Aside: we can see that the
joint probability formula
Now women receive 35%of all
on slide 24: promotions even though they are
P(A and B) = P(A) * P(B) just 15% of the workers.
only works if both events  No independence any longer!
A and B are independent
from one another!
0.20*0.15=0.03 ≠0.07
 Probability
31

 1.3 Probability Operators and Probability Rules

 The Complement Operator (“not”):

A

 The Intersection Operator (“and”)

A C B

 The Union Operator (“or”)

A B
 Probability
32

 1.3 Probability Operators and Probability Rules

 1. The Complement Rule
 An event and its complement event has to sum up to 1.
 Example:
 Say, A = “Being in New York City on Sunday”
 AC = “NOT being in New York City on Sunday”
The probability of “being in NYC” plus the probability of “not
being in NYC” has to add up to 100%.
 Thus:
AC

A
 Probability
33

 1.3 Probability Operators and Probability Rules

 2. The Addition Rule
 The probability that event A or event B (or both) occur:

To avoid to double-count
the overlapping part!
B (Because that overlap is part of both
A events A and B.)
 Probability
34

 1.3 Probability Operators and Probability Rules

 2. The Addition Rule for Mutually Exclusive Events
 Two events are said to be mutually exclusive, if both
cannot be true simultaneously.
 Example:
 Event A = “I am in New York.” A B

 Event B = “I am in Pittsburgh.”

 Addition Rule for Mutually Exclusive Events:

 Probability
35

 1.3 Probability Operators and Probability Rules

 3. The Multiplication Rule
 Used to compute the joined probability of 2 events. It is
based on the conditional probability formula which we have
explained earlier:

A C B

 Or, rewriting the last line:

 Probability
36

 1.3 Probability Operators and Probability Rules

 3. The Multiplication Rule for Independent Events
 We have seen earlier that for 2 independent events
we have:
and

 Therefore, when 2 events are independent:

B
 (that’s what we used on slide 23.)
 1

DISCRETE PROBABILITY
DISTRIBUTION
 Discrete Prob. Distribution
2

 2.1 Random Variables (RVs)

 An RV is a function that assigns a number to each outcome
of an experiment. (We usually use capital letters X,Y or Z
to refer to a RV.)

A discrete RV is one where the outcomes are countable.

An RV where the outcomes are uncountable is called
continuous.
 DiscreteRV: The sum of rolling 2 dice.
 Continuous RV: The time to finish a problem set.
 (for any 2 point of time there is a some point in time in between. )
 Discrete Prob. Distribution
3

 2.1 Random Variables (RVs)

 Example 1: Sum of 2 dice A random variable is a function or
rule that assigns a number to each
outcome in the sample space of an
1st die  experiment.
↓ 2nd die 1 2 3 4 5 6 Here:
“Rolling two ones”  X=2
1 2 3 4 5 6 7
“Rolling a 1 and 4”  X=5
2 3 4 5 6 7 8 “Rolling a 2 and 3”  X=5

3 4 5 6 7 8 9 In some experiments …
• the RV can take on the same
4 5 6 7 8 9 10 value for several outcomes
• the outcomes themselves are
5 6 7 8 9 10 11 numbers (e.g., return on an invest-
6 7 8 9 10 11 12 ment)  in those cases the value
of the RV simply is the numerical
event itself.
 Discrete Prob. Distribution
4

 2.1 Random Variables (RVs)

 Example 1: Sum of 2 dice

1st die  2: 1
↓ 2nd die 1 2 3 4 5 6 3: 2
1 2 3 4 5 6 7 4: 3
5: 4
2 3 4 5 6 7 8
6: 5
3 4 5 6 7 8 9 X= 7: 6
4 5 6 7 8 9 10 8: 5
9: 4
5 6 7 8 9 10 11
10:3
6 7 8 9 10 11 12 11:2
12:1

36 possible sum outcomes

 Discrete Prob. Distribution
5

 2.1 Random Variables (RVs)

 The probability of an RV’s outcome is written as P(X=x)
X # of P(x) or simply as P(x).
occurences
2 1 1/36
This is the probability distribution of the
3 2 2/36 discrete random variable X.
4 3 3/36
(We obtained in this example the probability
5 4 4/36 distribution by using the “classical approach”
6 5 5/36 to assign probabilities: assuming the dice are
fair, each outcome is equally likely.)
7 6 6/36
8 5 5/36
9 4 4/36
10 3 3/36  P(X=7) = P(7) = 6/36.
11 2 2/36
 P(X=12) = P(12) = 1/36.
12 1 1/36
Example:
SUM 36 1
 Discrete Prob. Distribution
6

 2.1 Random Variables (RVs)

 Example 2:
 LetX = the number of boys born to a family with 3 kids.
 Suppose getting a boy has a probability of 0.52.
  Find the RV and its probability distribution. Probability from
relative frequency
(Hint: A probability tree can help.) approach.
 Discrete Prob. Distribution
7

 2.1 Random Variables (RVs)

 Example 2:
 LetX = the number of boys born to a family with 3 kids.
 Suppose getting a boy has a probability of 0.52.
  Find the RV and its probability distribution. Probability from
relative frequency
(Hint: A probability tree can help.) approach.

Random Variable Probability distribution

0: 1 (GGG) 0: 1*0.48^3 = 11.1%
1: 3 (BGG, GBG, GGB) 1: 3*0.52^1*0.48^2= 35.9%
X= 2: 3 (BBG, BGB, GBB)
P(x) = 2: 3*0.52^2*0.48^1= 38.9%
3: 1*0.52^3 = 14.1%
3: 1 (BBB)
Sum: 100%
Number of branches * probability of
each branch in the tree.
 Discrete Prob. Distribution
8

 2.2 Probability Distributions

 Justas any other distribution, a probability distribution
can be displayed with a relative frequency bar chart
and be described via descriptive statistics.

Number of Boys in Family with 3 Kids

 Discrete Prob. Distribution
9

 2.2 Probability Distributions

 Just as any other distribution, a probability distribution
can be displayed with a relative frequency bar chart
and be described via descriptive statistics.
 The average of a probability

distribution is called the

“expected value”.

Number of Boys in Family with 3 Kids

Here: 0*0.111 + 1*0.359 +
2*0.389 + 3*0.141 = 1.56
 Discrete Prob. Distribution
10

 2.2 Probability Distributions

 The spread of a probability distribution is called
(just as before) variance and standard deviation.

Var (# of boys)=
(0-1.56)^2*0.11 +
(1-1.56)^2*0.359 +
(2-1.56)^2*0.389 +
(3-1.56)^2*0.141
= 0.745
Number of Boys in Family with 3 Kids
StdDev (# of boys) =
= sqrt(Var (#of boys) )
= sqrt(0.745) = 0.863
 Discrete Prob. Distribution
11

 2.2 Probability Distributions

 Laws of Expected Value

 The average of a constant is just the constant itself.

 Ifyou add to each value that the RV returns a constant, the

whole distribution just shifts. Hence the average itself will
also just be shifted by the constant c.

 Multiplying each value that the RV returns by a constant, will

simply shift the whole distribution (and increase its spread).
Hence the average itself will also just shift by that factor.
 Discrete Prob. Distribution
12

 2.2 Probability Distributions

 Laws of the Variance

 A constant does not have any variance/spread (by definition).

 If you add to each value that an RV returns a constant, the whole

distribution just shifts. The spread however remains unchanged.

 Multiplying each value in a distribution by a constant, will shift

the distribution by the factor to the left or right while increasing
its spread. As the variance is the avg. squared deviation from the
center, constant c becomes c2.
 Discrete Prob. Distribution
13

 2.2 Probability Distributions

 Examples 1 & 2:
 E(4X+5) = // let Y=4X and re-substitute it later
E(Y+5) = E(Y)+5 = E(4X)+5 = 4*E(X)+5

 V(2X+3) = // let Y=2X and re-substitute it later

V(Y+3) = V(Y) = V(2X) = 4*V(X)
 Discrete Prob. Distribution
14

 2.2 Probability Distributions

 Example 3:
 Suppose a slot machine costs $1 per game and works as
follows: with 30% probability you win $3, with 70%
probability you win nothing ($0). (In both cases you still had
to pay $1 to play the slot machine.)
 Q1: What is your expected payoff in the long run?
 Discrete Prob. Distribution
15

 2.2 Probability Distributions

 Example 3:
 Suppose a slot machine costs $1 per game and works as
follows: with 30% probability you win $3, with 70%
probability you win nothing ($0). (In both cases you still had
to pay $1 to play the slot machine.)
 Q1: What is your expected payoff in the long run?
 E(win of machine) = 0.3*$3 + 0.7*$0 = $0.90
 You still pay $1 to play the game  $0.90 -$1 = -$0.10
 In the long run you will lose on avg. 10 cents per game.
 Discrete Prob. Distribution
16

 2.2 Probability Distributions

 Example 4:
 Suppose that the return, r, of an asset can take one of three
values:
 r=4 with probability 0.2
 r=5 with probability 0.5
 r=6 with probability 0.3
 Calculate the expected value and the variance of the return of
the asset
 E[r]=0.2*4+0.5*5+0.3*6=5.1
 Var[r]=0.2*(4-5.1)^2+0.5*(5-5.1)^2+0.3*(6-5.1)^2
=0.2*1.21+0.5*0.01+0.3*0.81=0.242+0.005+0.243=0.49
 Discrete Prob. Distribution
17

 2.2 Probability Distributions

 Example 5:
 Suppose that the return, r, of an asset can take one of three
values:
 r=-2 with probability 0.3
 r=1 with probability 0.5
 r=5 with probability 0.2
 Calculate the expected value and the variance of the return
of the asset
 E[r]=0.3*(-2)+0.5*1+0.2*5=0,9
 Var[r]=0.3*(-2-0.9)^2+0.5*(1-0.9)^2+0.2*(5-0.9)^2
=0.3*8.41+0.5*0.01+0.2*16.81
=2.523+0.005+3.362=5.89