Stat test 3

Probability

a number between 0 →1 that quantifies the likelihood of an event occurring

Frequentist

the fraction of times something happens if you repeat the situation forever

Bayesian

the degree of belief, a number measuring how confident you are between 0 → 1

Probability calculation clasical

P(a)= (# of elements in A) / (# of elements in S)

Relative frequency (empirical method)

when outcomes are not equily likely. we estimate the probability from the data.

Probability calculation empirical method

P(a)= (#of times A occurred)/(# of trials)

Probability calculation empirical method (excel)

=COUNTIF(A2:A601, "Late")/COUNTA(A2:A601)

Subjective method

When there is no repeatable experiment and little or no data, a probability is assigned by reasoned judgment

Assigning Probabilities requirements

Every probability is between 0 and 1

The probabilities of all the outcomes in the sample space add up to 1

Mutually Exclusive

two events cannot happen at the same time

Union

“or” the probability of hears or Spades

requires addtion

Intersection

“and” the probability of hearts and spades

requires multipulcation

Complements P(A^c)

The remainder, everything thats not the specific scenario

Complements calculation

P(A^c)=1-P(A)

De Morgan’s Law

the complement of a union is intersection and the complement of an intersection is union

Addition Rule (mutually exclusive)

P(AuB)= P(a)+P(B)

Addition rule (not mutualy exclusive)

P(AuB)= P(a)+P(b)-P(A⋂B)

Multiplication rule (dependent events)

P(A⋂B)= P(a|B) *P(b)

Multiplication rule (indepent events)

P(A⋂B)= P(a) * P(b)

Independent events

two events are independent if the probability of occurrence by one event is not effected by the occurrence of the other event

Conditional probability

measures the likelihood of an event occurring given that another event has already happened.

Conditional probability calculation

P(A|B)= (P(A⋂B))/(P(B))

Law of total probability

P(a)= P(A⋂B)+ P(A⋂B^c)

or

P(a)= P(A|B)P(B)+P(A|B^c)P(B^c)

contingency table

Bayes Rule

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

counting rules (calculation)

p= (# elements in event)/(# elements in S)

Factorial

counts how many ways a group of objects can be organized

Fundamental counting principle

Ex: a restaurant offers 3 appetizers, 4 entrees, and 2 desserts. The number of different three course meals is 3×4×2=24

Factorial calculation

N! → x*(x-1)*(x-2)…1

combinations

counts the number of ways to choose x objects from a total of n objects

combinations formula (no repition)

c(n,x)= n!/((n-x)!x!)

combinations formula (with repition)

C(n+x-1,x)= (x+n-1)/(n-1)!x!

Permutation

also counts the number of ways to choose x objects from a total of n objects. However, the order in which the objects are listed does matter.

permutation calculation (no repetion)

P(n,x)= n!/(n-x)!

Permutation calculation (with repetition)

p(n,x)=n^x

Contingency table in excel

Pivot table, place one variable in the rows area, the other in the columns area, and any field in the values area as a count

Factorial in excel

=FACT(number)

Combinations in excel

=Combin(number, number_chosen) (no repetition)\

=Combina(number, number_chosen) (no repetition) with repition

Permutation in excel

=Permut(number,number_chosen) no repetition

=permutationa(number,number_chosen) with repetition

Expected Value

a measure of central location

Expected value calculation

E(x)= sum xf(x)

Variance

summarizes the deviation of the values of the random variable from the mean.

variance calculation

var(x)= E[(x-E(x))²)]=E(x²)-[E(x)]²

E(x²)- m²

uniform

is a probability distribution that assigns equal probability to each outcome

uniform calculation

f(x)=1/n

e(x)= (sum x)/n

var(x)= sum(x-m)²/n

Binomial distribution

counts the true # of successes in n trials of an experiment

x→ captures the # of successes

n→ number of trials

• every outcome can be classified as success or failure

• the probability of success is P and fixed

• trials are independent

binomial distribution calculation

f(x)= C_xⁿ (p^x)(1-p)^n-x

E(x)=np

var(x)=np(1-p)

hypergeometric distribution

a probability distribution that describes the outcome of drawing a sample from a population without replacement.

• trials are not indecent

• probability of success changes from trial to trial

hypergeometric calculation

Poisson Distribution

estimates the number of successes (x) over a specified interval of time or space

An experiment satisfies a Poisson process if:

• The number of successes with a specified time or space interval equals any integer between zero and infinity.

• The number of successes counted in non-overlapping intervals are independent.

• The probability of success in any interval is the same for all intervals of equal size and is proportional to the size of the interval.

poisson calculation

F(x)= (u^xe^-u)/x!

u- arival rate (average successes in the given time)

Binomial in excel

=binom.dist(x,n,p,FALSE)

poisson in excel

=Poisson.dist(x,mean, FALSE)

hypergeometric in excel

=hypgeom.dist(x,n,r,N,FALSE)

=HYPGEOM.DIST(sample_s, number_sample, population_s, number_pop, cumulative)

Probability mass function (PMF) in excel

set the last cumulative argument to FALSE

Cumulative probabilities in excel

set the last argument to TRUE

continuous random variables

a variable that can take on an uncountably infinite number of possible values within a given range or interval

uniform distribution

The uniform distribution assigns equal density across an interval [a,b]

the uniform distribution (calculation)

f(x)= 1/(b-a) (a≤x≤b)

e(x)= (a+b)/2

var(x)= (b-a)²/12

where a is the lower limit ans b us the upper limit f(x) 0 outside [a,b]

normal distribution

• It is symmetrical about the mean u.

• The mean is at the middle and divides the area of the distribution into halves.

• The total area under the curve is equal to 1.

• The distribution is completely determined by its mean and standard deviation.

The standard normal has a mean of 0 and a standard deviation of 1. Otherwise, it has the exact same properties as the normal distribution.

exponential distribution

is useful in computing probabilities for the time it takes to complete a task. It describes the time between events in a Poisson process

Exponential distribution calculation

f(x)= (1/u)e^-x/u

Triangle distribution

is characterized by a single mode (the peak of the distribution) and two boundaries. It is often used in situations where the lower and upper bounds of a potential outcome are known, but the exact likelihood of the outcome is uncertain.

Triangular distribution formula

Normal distribution (excel)

Norm.dist(x,mean,standard_dev,cumulative)

set cumulative to TRUE for P(X≤x)

set cumulative to FALSE for the density

for standard normal (u=0,ō=1) use Norm.s.dist(z, cumulative)

to find value given a probability use Norm.inv(probability,mean,standard_dev) or Norm.S.INV(Probability)

Exponential excel

=expon.dist(x,1/u, cumulative)