Probability

Question 1

What is a random experiment?

Answer 1

a process that results in a number of possible outcomes, none of which can be predicted with certainty;

Question 2

What is the sample space?

Answer 2

sample space of a random experiment is a list of all possible outcomes

E.g. roll a die: sample space:

        S={1, 2, 3, 4, 5, 6}.

Question 3

How can variance and standard deviation be used in real life?

Answer 3

Higher standard deviation/variance = higher return

Question 4

What is a mutually exclusive outcome?

Answer 4

When two events cannot occur at the same time in a single trial

Therefore P(A) x P(B) does not equal to P(AxB)

Question 5

What is a collectively exhaustive outcome?

Answer 5

one of the events must occur AKA one or the other

eg. heads and tails in a coin toss is collectively exhaustive because one of them must occur. If head does not occur, tails must occur

Question 6

What is the probability of sample space?

Answer 6

P(S) =1

Question 7

What is P(A|B)?

Answer 7

—Conditional probability that A occurs, given that B has occurred:

Question 8

What is P(A or B) = P(A U B) = P(A union with B)?

Answer 8

A occurs, or B occurs, or both occur

P(A or B) = P(A) or P(B) - P(A and B)

Question 9

What is P(A and B) = P(A ∩ B) = P(A intersection with B) ?

Answer 9

A and B both occur

Question 10

What is another way to express P(A)?

Answer 10

P(A) = P(A∩B) + P(A ∩Bc)

Question 11

What is another way to express P(A or B)?

Answer 11

P(A or B)=P(A)+P(B)-P(A and B)

Question 12

How to test if two events are independent?

Answer 12

P(A|B)=P(A) or P(B|A)=P(B)

P(A and B) = P(A)*P(B)

Question 13

Prove that P(A|B) = P(A)

Answer 13

= [P(A and B)]/P(B)

=[P(A)*P(B)] /P(B)

=P(A)

Question 14

what is marginal probability

Answer 14

—Computed by adding across rows or down columns - calculated in the margins of the table

Question 15

Prove the complement rule

Answer 15

—Given an event A and its complement, Ā, so that A+ Ā=S;

Know that P(S)=1;

so P(A)+P(Ā)=1;

therefore P(Ā)=1-P(A)

Question 16

Another way to express P(A|B)

another way to express P(B|A)

Answer 16

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

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

Question 17

another way to express P(A and B)

prove

Answer 17

as P(A|B) = P(A and B) / P(B)

P(A and B) = P(A|B) x P(B)

Question 18

formula for expected value

Answer 18

Question 19

random variable example

Answer 19

—Imagine tossing three unbiased coins.

S= {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT)

—8 equally likely outcomes.
—Let X = number of heads that occur.
—X can take values 0, 1, 2, 3.

Question 20

how to denote random variable and actual realised values?

Answer 20

—Denote random variables (X, Y, etc) in upper case
—Denote actual realised values (x, y etc) in lower case

Example: X is the random variable that can take values 0, 1, 2, 3.

Actually perform experiment, find the pattern HTT. Then x=1.

Question 21

what is a discrete random variable?

Answer 21

—discrete random variable has a countable number of possible values, e.g. number of heads, number of sales etc.

Does not have to be finite; but values can be strictly ordered.
based on counting process

Question 22

what is a continuous variable?

Answer 22

continuous random variable has an infinite number of possible values

based on measuring process

Question 23

what are features of probability distribution?

Answer 23

Sum of proabilities must equal 1

P(X=x)

Question 24

what are frequentist probabilities?

Answer 24

—Probabilities associated with r.v.s are often associated with relative frequencies.
E.g. interested in number of photocopiers sold by different firms. Estimate probabilities from frequencies

Question 25

what is the probability disturibution for a discrete variable?

Answer 25

mutually exclusive list of all possible numerical outcomes along with probability of occurence of each outcome

Question 26

what are numerical variables? how can they be further categorised?

Answer 26

numerical variables produce numerical repsonses further categorised into discrete and continuous numerical variables

Question 27

What are rules for expectations? given that c is a constant

Answer 27

—E(c)=c —E(cX)=cE(X) —E(X-Y)=E(X)-E(Y) —E(X+Y)=E(X)+E(Y) —E(XY)=E(X)\*E(Y) only if X and Y are independent

Question 28

2.Let Z=3X+2Y-2XY+3, with E(X)=3, E(Y)=5, X and Y independent calculate Z

Answer 28

= E(3X+2Y-2XY+3) =3E(X)+2E(Y)-2E(X)\*E(Y)+3 =3\*3+2\*5-2\*3\*5+3 =9+10-30+3 = -8

Question 29

what does variance measure?

Answer 29

—spread/dispersion of distribution

Question 30

what are laws for variances?

Answer 30

1. V(c)=0 2. V(cX)=c²V(X) 3. V(X+c)=V(X) 4. V(X+Y)=V(X)+V(Y) if X and Y are independent 5. V(X-Y)=V(X)-V(Y) if X and Y are independent

Question 31

a) Find the mean and variance of X b) Find the mean and variance of Y = 2\*X+5

Answer 31

Question 32

what is bivariate distribution? distnguish with univariate distribution

Answer 32

Distribution of a single variable – univariate —Distribution of two variables together – bivariate —So, if X and Y are discrete random variables, then we say p(x,y) = P(X=x and Y=y) is the joint probability that X=x and Y=y.

Question 33

show a bivariate distribution table

Answer 33

—Let X be the number of heads. Let Y be the number of changes of sequence, i.e. the number of times we change from H →T or T→H

Question 34

how can you tell if random variables are independent?

Answer 34

—If the random variables X and Y are independent, then ## Footnote P(X=x and Y=y) = P(X=x) \* P(Y=y) p(x,y) = p_x(x) \* p_y(y)

Question 35

how do you calculate sum of two random variables?

Answer 35

by using bivariate distribution

Question 36

—Then, P(X+Y=2) = sum of all joint probabilities for which x+y=2; Continue to do for P(X+Y=3), P(X+Y=4) Show

Answer 36

P(X+Y=2)—= p(0,2) + p(1,1) + p(2,0) ## Footnote = 0.07 + 0.06 + 0.06 =0.19 ...continue for P(X+Y=3), P(X+Y=4)

Question 37

evaluate mean and variance of (X+Y)

Answer 37

E(X+Y) = 1.2 V(X+Y) = 0.56

Question 38

what happens when joint distribution of X and Y is unknown?

Answer 38

—use the rules learnt earlier: ## Footnote If a and b are constants, then E(aX+bY) = aE(X) + bE(Y) V(aX+bY) = a²V(X) + b²V(Y) – only if X and Y are independent!

Question 39

—E(X) = µx and E(Y)= µy, then the covariance between X and Y is given by

Answer 39

Question 40

what is the formula for variance when there is correlation?

Answer 40

Question 41

how does correlation relate to investment risk?

Answer 41

—Analysts reduce risk by diversifying their investments – that is, combining investments where the correlation is small.

Question 42

—An investor forms a portfolio by putting 25% of his money in Stock A and 75% in stock B, with parameters below. what is the expected return of the portfolio?

Answer 42

—E(RA)=0.08, E(RB)=0.15 —Rp=0.25\*RA+0.75\*RB —E(Rp) = 0.25\*E(RA)+0.75E(RB) = 0.25\*0.08+0.75\*0.15 = 0.1325 —Expected return of the portfolio is 13.25%.

Question 43

Determine the variance of the return if ρ=1

Answer 43

Question 44

Counting rule 2: what do you do when you have there are k1 events on the first trial, k2 events on the second trial…and kn events on the nth trial then the possible outcomes

Answer 44

(k1)(k2)…(kn)

Question 45

Counting rule 3: how do arrange number of n items in order?

Answer 45

N!

Question 46

Counting rule 4: when an entire subject of an entire group of items needs to be arranged in order

Answer 46

Permutation : n!/ (n-x)!

Question 47

Counting rule 5: select subset from entire group of items without the need for order

Answer 47

Combination: n! (n-x)! x!

Question 48

What is a Bernoulli process?

Answer 48

Two possible outcomes for each trial, labelled success and failure Probability of success is p, probability of failure is (1-p) Trials are independent – the outcome of one trial does not affect the outcomes of any other trials.

Question 49

What properties does a binomial experiment have?

Answer 49

1. A fixed number of trials, n 2. Two possible outcomes for each trial, labelled success and failure 3. Probability of success is p, probability of failure i 4. Trials are independent – the outcome of one trial does not affect the outcomes of any other trials.

Question 50

What is the binomial random variable?

Answer 50

binomial random variable is defined as the number of “successes” in the n trials

Question 51

How to express the probability that we will have k successes?

Answer 51

X~Bin(n=….,p…..) Eg. n =12 p =3 Then X~Bin(n=12,p=0.25)

Question 52

Show mean and variance of binomial distribution

Answer 52

Question 53

guideline for using binomial table

Answer 53

—Note that P(X ≤0)= P(X=0) —P(X ≤k) →use tables —P(X=k) = P(X≤k) – P(X≤(k-1)) —P(X —P(X≥k) = 1- P(X≤[k-1]) —P(X\>k) = 1- P(X ≤k)

Question 54

features of bivariate distributions

Answer 54

—Marginal distributions —Independence —Sum of two random variables —Covariance and Correlation —Linear Combinations of Variables

Question 55

features of binomial distribution

Answer 55

—binomial experiment bernoulli trial binomial formula mean and variance of a binomial variable binomial tables.

Question 56

what does expected value show?

Answer 56

long term average of discrete random variable

Question 57

formula for variance

Answer 57

E(X²) - [E(X)]²

Question 58

what are marginal distribution functions?

Answer 58

Question 59

In the casino game roulette, a gambler can bet on which of 38 numbers will be the result when the roulette wheel is spun. On a $2 bet, a gambler gains $70 if he or she picks the right number, but loses the $2 otherwise. a. Let X = amount gained or lost on a $2 bet on a roulette number. Write out the probability distribution of X.

Answer 59

Question 60

In the casino game roulette, a gambler can bet on which of 38 numbers will be the result when the roulette wheel is spun. On a $2 bet, a gambler gains $70 if he or she picks the right number, but loses the $2 otherwise Imagine a player decides to bet on the same number for 10 trials in a row, and count Y = the number of wins in those 10 trials. What sort of distribution does Y have? What are the mean and variance of this distribution?

Answer 60

Hence, Y will be a binomial variable. Y~Bin (n=10, p=1/38) ``` E(Y) = np = 10/38 Var(Y) = np(1-p) =10\*1/38\*37/38=370/1444= 185/722 ```

Question 61

In the casino game roulette, a gambler can bet on which of 38 numbers will be the result when the roulette wheel is spun. On a $2 bet, a gambler gains $70 if he or she picks the right number, but loses the $2 otherwise . Again considering Y, the number of wins in 10 trials, what is the probability that the player wins on at least 5 trials? What is the probability that there are no wins in the 10 trials?

Answer 61

``` P(Y≥5) = 1-P(Y≤4) = 1- 1.0000 = 0 P(Y=0) = 0.765916 ``` ¹⁰C₀ x (1/38)⁰x (37/38)¹⁰

Question 63

Answer 63