stats exam prep. dr. lin lin. warning the goal of this workshop is to go over some basic concepts in...

Stats Exam Prep.

Dr. Lin Lin

WARNING

• The goal of this workshop is to go over some basic concepts in probability and statistic theories required for IS 665

• It is NOT to help you pass the exam

• NO EXAM QUESTIONS will be covered here

Probability

• Probability is the measure of how likely something will occur.

• It is the ratio of desired outcomes to total outcomes. – P(event) = (# desired events) / (# total events)

• Probabilities of all outcomes sums to 1.

BEFORE Probability

• We need to learn to count the number of possible events

• Exercise I: How many different five-digit numbers exist?

• How did you get the answer?

BEFORE Probability

• Exercise II: How many different five-digit numbers WITHOUT 0 exist?

• How did you get the answer?

BEFORE Probability

• Exercise III: US phone number is in the format of (###) – ### - ####

– The first digit cannot be zero– There cannot be a “000-0000” number– How many numbers are possible?

• How did you get the answer?

BEFORE Probability

• Exercise IV: let’s make a three-digit number. There is only one rule: no two digits could be identical. How many numbers could we make?

• How did you get the answer?

BEFORE Probability

• Exercise V:

• You could rearrange these shapes anyway you want. However, cannot be on either side. How many different ways could we have?

• How did you get the answer?

Probability Example

• If I roll a number cube, there are six total possibilities. (1,2,3,4,5,6)

• Each possibility only has one outcome, so each has a PROBABILITY of 1/6.

• For instance, the probability I roll a 2 is 1/6, since there is only a single 2 on the number cube.

Practice

• If I flip a coin, what is the probability I get heads?

• What is the probability I get tails?

• Remember the equation? Number of desired outcomes divided by number of possible outcomes

Answer

• P(heads) = 1/2• P(tails) = 1/2

• If you add these two up, you will get 1, which means the answers are probably right.

Answer

• Let’s make it harder – assuming that the coin is not fair, and P(H) = 0.6

• What is the chance of getting a tail in one flip?

Bernoulli Trial

• In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted.

• So it is basically coin-flipping with a p of not necessarily 0.5

Two or more independent events

• If there are two or more independent events, you need to consider if it is happening at the same time (and) or one after the other (or).

And

• If the two events are happening at the same time, you need to multiply the two probabilities together. This probability is called joint probability

• Usually, the questions use the word “and” when describing the outcomes.

• P(A & B) = P(A)*P(B)

Joint Probability

Just a fancy way of saying “AND”◦ p(I will listen to Backstreet Boys Today) = 0.8◦ p(I will eat at Subway today) = 0.7

What is the probability that I will listen to Backstreet Boys AND eat at Subway?◦ 0.7 * 0.8 = 0.56?

WHEN EVENT A AND B ARE INDEPENDENT:◦ P(A&B) = P(A) * P(B)

Or

• If the two events are happening one after the other, you need to add the two probabilities.

• Usually, the questions use the word “or” when describing the outcomes.

• P(A or B) = P(A) + P(B)

Practice

• If I roll a number cube and flip a coin:

– What is the probability I will get a heads and a 6?– What is the probability I will get a tails or a 3?

• How did you get them?

Answers

• P(heads and 6) = 1/2 x 1/6 =1/12

• P(tails or a 5) = 1/2 + 1/6 = 8/12 = 2/3

Summary: Independent Events• One event has no influence on the outcome of

another event

• If events A & B are independentthen P(A&B) = P(A)*P(B)P(A or B) = P(A) + P(B)

• Coin flippingif P(H) = P(T) = .5 thenP(HTHTH) = P(HHHHH) = .5*.5*.5*.5*.5 = .55 = .03

Summary: Independent Events

• Coin flippingif P(H) = P(T) = .5 thenP(HTHTH) = P(HHHHH) = .5*.5*.5*.5*.5 = .55 = .03

• What if P(H) = 0.6 (Bernoulli trial)? – What is P(HTHH)?– What is P(at least one head in five trials)?

• if you are flipping a coin and it has already come up heads 6 times in a row, what are the odds of an 7th head?

.5

• Is P(10H) = P(4H,6T)?

Next year the economy will experience one of three states: a downturn, stable state, or growth. The following probability matrix displays joint probabilities of a bond default and the economic state:

For example, the joint probability that the economy is stable and the bond defaults is 1.0%; the unconditional probability that the economy will be stable is 50.0% = 49.0% + 1.0%.

What is the chance of the bond default?

What is the chance of an economy downturn?

Next year the economy will experience one of three states: a downturn, stable state, or growth. The following probability matrix displays joint probabilities of a bond default and the economic state:

For example, the joint probability that the economy is stable and the bond defaults is 1.0%; the unconditional probability that the economy will be stable is 50.0% = 49.0% + 1.0%.

Knowing that the bond survived,

what is the chance that economy is stable?

Conditional Probability

• Concern the odds of one event occurring, given that another event has occurred

• P(A|B)=Prob. of A, given B

Examples

• P(Professor Lin walks in without a Pepsi in his hand) = 0.1

• HOWEVER…• P(Professor Lin walks in without a Pepsi in his hand | you

promise to give me $1,000,000 if I do so) = 1 !!

• What changed my behavior?

• P(B|A) = P(A&B)/P(A)

• if A and B are independent, thenP(B|A) = P(A)*P(B)/P(A) = P(B)

Conditional Probability (cont.)

The Chain Rule What if A and B ARE dependent of each other?

◦ p (I am teaching IS 665 today) = 1/7◦ p (I am eating at Subway today) = 0.7

What is the chance that I am teaching 665 today and eating at Subway?◦ p (I am teaching IS 665 today & I am eating at Subway today) = 0!

WHY?◦ Because to teach 665, I have NO TIME to eat at Subway!◦ In other words, these two events are dependent

The Chain Rule What is the chance that I am teaching 665 today and eating

at Subway?◦ p (I am teaching IS 665 today & I am eating at Subway today) =

p (I am eating at Subway today | I am teaching 665) * p (I am teaching 665) = 0 * 1/7 = 0

To put it (semi) formally:◦ P(A & B) = P (A | B) * P (B) = P(B | A) * P(A)

The Bayes Rule The Chain Rule Shows us:

◦ P(A & B) = P (A | B) * P (B) = P(B | A) * P(A)

P (A | B) = P(B | A) * P(A) / P(B) !!!

This is the Bayes Rule

The Bayes Rule

P(B | A) = P(B) * P (A | B) / P (A)

BAPBPBAPBP

BAPBPABP

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Exercise

If we observe that the bond has defaulted, what is the (posterior) probability that the economy experienced a downturn?

a. 0.60%b. 19.40%c. 26.33%d. 31.58%

Exercise

If it is snowing, there is a 80% chance that class will be canceled. If it is not snowing, there is a 95% chance that class will go on. Generally, there is a 5% chance that it snows in NJ in the winter.

If we are having class today, what is the chance that it is snowing?

BAPBPBAPBP

BAPBPABP

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Distribution

• How to Read a Histogram

Normal Distribution

• Watch the demo

Regression• Predict the target value with the attributes by a function:

• Handle numeric attributes and predict numeric value.

Regression

• Goal: minimize the error • An example

Variable Estimate Std. Error t value Pr(>|t|)

(Intercept) 5000 XXX XXX 0.020

edu_level 1000 XXX XXX 0.001

IQ 50 XXX XXX 0.814

experience 300 XXX XXX 0.004

gender -2000 XXX XXX 0.300

What is Regression, anyway?

Number of nights I illegally parked Chance that I will get a ticket

0 3

1 21

2 36

3 44

4 66

5 81

y = 15.229x + 3.7619

InterceptCoefficient

If I parked illegally 6 nights in a row, how likely am I to get a ticket?

P value

What is Regression, anyway?

• You now know how to interpret a regression model

• But how do we build one?– That will be covered in IS 665