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 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 It is NOT to help you pass the exam NO EXAM QUESTIONS will be covered here NO EXAM QUESTIONS will be covered here

Probability how likely 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 We need to learn to count the number of possible events Exercise I: How many different five-digit numbers exist? Exercise I: How many different five-digit numbers exist? How did you get the answer? How did you get the answer?

BEFORE Probability Exercise II: How many different five-digit numbers WITHOUT 0 exist? Exercise II: How many different five-digit numbers WITHOUT 0 exist? How did you get the answer? How did you get the answer?

BEFORE Probability Exercise III: US phone number is in the format of Exercise III: US phone number is in the format of (###) – ### - #### – The first digit cannot be zero – There cannot be a “ ” number – How many numbers are possible? How did you get the answer? 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? 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? How did you get the answer?

BEFORE Probability Exercise V: Exercise V: You could rearrange these shapes anyway you want. However, You could rearrange these shapes anyway you want. However, cannot be on either side. How many different ways could cannot be on either side. How many different ways could we have? we have? How did you get the answer? 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) 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. 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. For instance, the probability I roll a 2 is 1/6, since there is only a single 2 on the number cube.

Practice heads If I flip a coin, what is the probability I get heads? tails What is the probability I get tails? Number of desired outcomesnumber of possible outcomes 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

independent 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 multiply joint probability 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” 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? 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: WHEN EVENT A AND B ARE INDEPENDENT: ◦ P(A&B) = P(A) * P(B)

Or add 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 independent then P(A&B) = P(A)*P(B) P(A or B) = P(A) + P(B) Coin flipping if P(H) = P(T) =.5 then P(HTHTH) = P(HHHHH) =.5*.5*.5*.5*.5 =.5 5 =.03

Summary: Independent Events Coin flipping if P(H) = P(T) =.5 then P(HTHTH) = P(HHHHH) =.5*.5*.5*.5*.5 =.5 5 =.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 7 th 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, then P(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? 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? 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? 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? 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: 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: 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)

Exercise If we observe that the bond has defaulted, what is the (posterior) probability that the economy experienced a downturn? a. 0.60% b % c % d %

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?

Distribution How to Read a Histogram

Normal Distribution Watch the demo

Regression

Regression VariableEstimateStd. Errort valuePr(>|t|) (Intercept)5000XXX edu_level1000XXX IQ50XXX experience300XXX gender-2000XXX 0.300

What is Regression, anyway? Number of nights I illegally parkedChance that I will get a ticket y = x InterceptCoefficient If I parked illegally 6 nights in a row, how likely am I to get a ticket?

P value P value

What is Regression, anyway? You now know how to interpret a regression model You now know how to interpret a regression model But how do we build one? But how do we build one? – That will be covered in IS 665