Intro to Probability

key aspect of solving real business problem

dealing with uncertainly

sources of uncertainty

• demand for products

• stock price returns

• time between arrivals

• daily sales

random variables

the name of uncertainty in numerical quantity in probability and statistics

probability

a way to deal with and manage uncertainty , an intuitive concept

experiment

any process whose outcome is not known in advance

sample space

the set of all possible outcomes of an experiment

event

a set of possible outcomes of an experiment

probability

the chance that an event will occur

properties of porbability

• all probabilities are between 0 and 1

• the closer of an event is to 1, the more likely it is to occur

• if event a is a subset of event b, then the probability of b is greater than or equal to probability of a

  • event a = sunny

  • event b = sunny, cloudy

  • P(b) greater than equal to P(a)

• rule of complements: P(a^c) = 1 - P(a)

  • a^c is the event that event a doesn’t occur

• addition rule

• conditional probability and multiplication rule

• probabilistic independence

• law of total probability

random variable

a variable that can take on different values, each with an associated probability

will hold the outcome of a random event

probability distribution

describes the likelihood of a random variable

• its a list of possible outcomes (sample space/population) and their associated probabilities

• population probability is made up w assumptions or estimate it using historical data

discrete random variable

take on only a countable number of distinct values

the probability mass function, pmf

X = random variable, x = fixed outcome

expected value:

• is the weighted average of all the k values, with their probabilities serving as weights

• E[.] is a function we apply to random variables

continuous random variable

takes on an infinite number of real values in some sane

the probability density function f(x), PDF, is the likelihood of x. CDF is denoted as F(x). the area under f(x) within a given interval

x is a continuous RV, then P(X=x)=0, for every real number x

cumulative probability distribution function, cdf

describes the probability that the random variable is less that or equal to some value

continuous random variable

the probability density function f(x) (PDF), is the likelihood of x.

CDF = F(x)

expected value of a discrete random variable

the weighted average of all the k values, with their probabilities serving as weights

E[.] is a function applied to random variables

expected value

long run average

variance of a random variable

quantify the variability of a random variable. variance measures the variability of the outcomes of X around its expected values (mean) X

Var(.) like E[.]

correlation of two random variables

independence of two events

intuition: the occurrence of one event (A) doesn’t affect the probability of occurrence of the other event (B)

P(A|B) = P(A)

P(B|A) = P(B)

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

if two events are independent

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

variance of sum of random variables if variables are independent

Var(X+Y) = Var(X)+Var(Y)

variance of sum of random variables if variables are dependent

Var(X+Y) = Var(X)+Var(Y)+2Cov(X,Y)