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Question 1

Associative Law for Union and Intersection

Answer 1

(EUF)UG = EU(FUG)

E or F or G = E or F or G

EF(G) = (E)FG

E and F and G = E and F and G

Question 2

Chubyshovs inequality

Answer 2

If we are trying to identify how much of a dataset lies between the values of x̄ +-ks where s = standard deviation and k = some number then

% min = 100(1- 1/(k²))

k>1 otherwise the probability = 0

Question 3

Communitive law for union/intersection

Answer 3

E U F = F U E

Event E or F = Event F or E

EF=FE

Event E and F = Events F and E

Question 4

Compliment

Answer 4

If we have an event, E, within the sample space, S then E^c = compliment of E and includes everything that is not E

Question 5

Correlation Coefficient

Answer 5

r = [Σ (x_i - x̄)( y_i - ȳ)]/(n-1)s_xs_y = [Σ (x_i - x̄)( y_i - ȳ)]/[Σ (x_i - x̄)²( y_i - ȳ)²]^.5

This says that if we have a paired dataset such that x_i,y_i are the pairs and are described by their respective means such that y = mx + b then this statistic will indicate the linearity of the pairs of data

Question 6

Cumulative Frequency

Answer 6

This shows the bins as a function of an additive frequency.

These are also called Ogives

Question 7

Demorgans Laws

Answer 7

(EUF)^C = E^CF^C

E or F do not occur = E not occurring and F not occurring

(EF)^C = E^C U F^C

E and F not occurring = E not occurring or F not occurring

Question 8

Gini Coefficient

Answer 8

The gini coefficient (G) is the integral of the area between L(p) = 1 and the Lorenz Curve. It has a maximum value of .5 and a minimum value of 0

G=1-2B where B = area under Lorenze curve, L(p)

Question 9

How to use the weighted probability of E?

Answer 9

If tasked with finding P(F|E) where E is the second event

Then P(F|E) = P(FE)/(P(E)

where P(FE) = P(F)P(E|F) and P(E) = P(F)P(E|F) + P(F^c)P(E|F^c)

Question 10

Independent events

Answer 10

If P(E|F) = P(E) then E and F are independent and E is not a function of event F

and P(EFG)=P(E)P(F)P(G)

These occur when two events occur at the same time or in distinct independent localles.

Ex: Covid cases in Mexico and Greenland on some day, two balls being selected from a jar at once (if not at the same time their chances are not independent)

Question 11

Lorenz Curve

Answer 11

This is a cumulative curve showing the income distribution

Question 12

mean

Answer 12

x bar = Σx/n = Σ v*f/n

where v = bin value and f = frequency

Question 13

Mean vs. Median when to use

Answer 13

Generally mean gives a better understanding of the dataset in terms of describing the data. The median should be used when probabilities are involved and/or the value is being used to understand the order of a group.

Ex: Housing. The mean income would be best for determining what the average person in an area can spend on a home but if we want to design housing where we could expect 50% of the population could live (P(Affordable)=.5) then the median is more useful.

Question 14

Median

Answer 14

This is the middle value of a sample when data is arranged from least to greatest

If n is odd then the median value occurs at n = (n+1)/2

If n is even then the median is the average of (n/2)+1 and n/2

Question 15

number of unique groups in a set

Answer 15

If we have a collection of n = # of objects and we want to know how many unique combinations of size r can be made when order matters

= n!/[(n-r)!r!]

This says that for a sample size=n that we can arrange “r” elements this many ways uniquely. If we have a sample space this will define the number of potential outcomes that are possible.

If we want to know the possibility of a subset occurance within a group this will be given by (# of combinations in subset 1)*(#comb in 2)/(total # comb)

Question 16

P(E) = ? as a weighted average

Answer 16

P(E) = P(E|F)P(F) + P(E|F^c)(1-P(F))

The P(E) = The weighted average of E occurring if F has occurred and if E occurs and F does not occur

Where E occurs as the consequence of F.

Question 17

P(E) Expansion using compliments

Answer 17

P(E) = P(EF) + P(EF^C))

Probability of E = Prob of E and F + Probability of E and not F

Question 18

P(E|F) = ?

Answer 18

P(E|F) = P(EF)/(P(F)

Probability of E occurring given F has occurred = the probability E and F occur divided by the probability F occurs

Question 19

P(E|F^c) = ? (expand)

Answer 19

P(E|F^C) = P(EF^C)/P(F^c)

This is says that the probability of E occurring given that F does NOT occur equals the probability of E occurring and F not occurring divided by the probability F does not occur.

Question 20

Permutations

Answer 20

This is a specific arrangement of a set of objects where the total number of permutations available to a subset of things is equal to n! where n is the total number of things in the subset

Question 21

r meaning

Answer 21

If the slope relating y and x is <0 then r <0 and vice versa. the absolute value of r indicates the linearity of the relationship

If r is for (x, y) where w = a + bx and z = c + dy then

r(x,y) = r(w,z)

Question 22

Sample 100p percentile

Answer 22

The data point equal to where less than 100*p% of data lies. It includes that data point

p=probability as a decimal

Question 23

Sample Space

Answer 23

S = sample space = all possible outcomes to some experiment. This can be both discrete or nominal data. The subset of data is the event

ex: an experiment predicting the gender of children

S = {g,b} and E={g} F={b}

Question 24

Sample spaces with equally likely outcomes

Answer 24

This refers to a sample space where each outcome has an equal probability of occurring, aka there is no weight to a particular outcome

In this scenario P(E) = 1/N = p

These sample spaces have a total number of outcomes given by n! where n is the number of objects in the sample space and n! gives the total number of unique combinations of these objects. If there is a number of experiments, m, each with n number of outcomes then the total number is m*n

Question 25

Three axioms of Probability

Answer 25

0

2: P(S) = 1
3: P(U_iⁿ E_i) =Σ_iⁿ P(E_i) = P(E₁) + P(E₂) +… +P(E_n)

This is assuming that E_i and E_i+1 are mutually exclusive and says that if this is true that the probability of one of the events occurring is equal to the summation of each individual events.

Question 26

Union

Answer 26

E U F = E or F which means that any outcomes within either subset or event E or F are valid.

Question 27

When is conditional probability particularly useful?

Answer 27

It is used when there is limited information within a problem (you are attempting to derive the probability of an event based on other events)

or

It is the easiest way to find the probability of a cause or input to an event with new information (backwards reasoning)

Question 28

P(E or E^c) = ?

Answer 28

P(E or E^c) = P(S) = 1

Question 29

P(E or F) if E and find are not mutually exclusive

Answer 29

P(E or F) = P(E) + P(F) -P(EF)

Question 30

Odds of an event

Answer 30

The odds of an event occuring is the ratio of the event occuring to not occuring

odds = P(E)/P(E^c)

Question 31

Mass density function

Answer 31

This is the probability of a function being equal to or less than a given value hence it is the integral from the lower bound to some value.

Question 32

density function

Answer 32

this is a function which describes the probability of a random variable equalling a specified value. The integral of the density function is the mass function.

Question 33

If given a joint density function and asked to find P(x>y) what is the procedure?

Answer 33

the P(x>y)=∫ ∫ ₀^x f dy dx

This says that the probability of y being less than x is equal to the mass probability function of y=x where the bounds on x are negative and positive infinity

Question 34

Procedure for finding the joint probability function for discrete events?

Answer 34

1. ID variables
2. ID if independent (if independent then the probabilities can be separately evaluated and then multiplied using “and” statements)
3. Logic through the first few probabilities
4. Try to identify a pattern that can be used for combination/permutation notation
5. If possible create an equation. If not, work through each probability.

Question 35

How to use combinations to find the discrete chance of an event occurring?

Answer 35

P of an event occuring is equal to the combination of the number of variables within the scenerio into the number of anticipated responses. This is divided by the total number of options per event.

In other words this is the total number of unique combinations of events divided by the number of those that meet the criterion divided by total events. This results in the number of events that meet the criterion specific over the total number of events or the probability.

Question 36

Random Variable

Answer 36

A random variable is a variable that does not describe a discrete event but the meaning of an event. It is a numerical value which is derived from the result of an experiment.

Question 37

Expectation formula

Answer 37

The E[X] = ∫ P(X=x)*X

This is the weighted average of the variable’s values