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Why Probability? Probability theory describes the likelihood of observing various outcomes for a given population Statistics uses rules of probability as a tool for making inferences about or describing a population using data from a sample Some Concepts

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why probability
Why Probability?
  • Probability theory describes the likelihood of observing various outcomes for a given population
  • Statistics uses rules of probability as a tool for making inferences about or describing a population using data from a sample
some concepts
Some Concepts
  • Experiment: the process by which an observation is obtained
    • e.g. the roll of a dice
  • Event: the outcome of an experiment.
    • e.g. observe a 1; observe an odd number (1,3,5)
  • Simple Event: an event that cannot be decomposed
  • Sample Space (S): set of all events
definition of probability
Definition of Probability
  • The probability of an event A is a measure of our belief that the experiment will result in event A.
    • If we repeat an experiment N times, and event A occurs n times, then P(A) = n / N
  • Computing probabilities in this way is infeasible in practice, but implications are useful
probability rules
Probability Rules

For an event A:

  • P(A) is between 0 and 1, inclusive
  • If A contains t simple events, then

P(A) = P(E1) + P(E2) + … + P(Et)

For a sample space S with s simple events:

P(S) = P(E1) + P(E2) + … + P(Es) = 1

event composition
Event Composition
  • The Intersection of events A and B is the event that both A and B occur
    • denoted AB or A∩B
  • The Union of events A and B is the event that A or B or both occur
    • denoted AUB
event composition6
Event Composition
  • A and B are mutually exclusive if there are no simple events in A∩B. If A,B are mutually exclusive then:

(1) P(A∩B) = 0

(2) P(AUB) = P(A) + P(B)

  • The complement of an event A consists of all simple events that are not in A
    • denoted
conditional probability
Conditional Probability
  • In some cases events are related, so that if we know event A has occurred then we learn more about an event B
  • Example: Roll a die

A: observe an even number (2,4,6)

B: observe a number less than 4 (1,2,3)

if we know nothing else then P(B) = 3/6 = 1/2

But if we know A has occurred then P(B | A) = 1/3

conditional probability8
Conditional Probability
  • More generally, we can express the conditional probability of B given that A has occurred as:
  • We can rewrite this formula to get the Multiplicative Rule of Probability:
independence
Independence
  • Events are not always be related. Events A and B are independent if and only if:
  • If A and B are independent, then from Multiplicative Rule of Probability:
rules of probability
Rules of Probability

Given 2 events A and B:

  • Additive Probability:
    • If A and B are mutually exclusive then
      • P(AB)=0
      • P(A+B) = P ( A )+ P ( B )
  • Total Probability:

for mutually exclusive B1, B2, …

bayes rule
Bayes Rule
  • Take into account prior information when computing probabilities
  • Let S1, S2, S3,…Sk represent k mutually exclusive, only possible states of nature with prior probabilities P(S1), P(S2),…P(Sk). If an event A occurs, the posterior probability of Si given A is the conditional probability
random variables
Random Variables
  • X is a random variable if value that it assumes depends on the random outcome of an experiment
  • A random variable may be
    • Discrete: countable number of values
    • Continuous: infinite number of values
discrete probability distribution
Discrete Probability Distribution
  • The probability distribution for a discrete random variable is a formula, table or graph that provides p(x), the probability associated with observing x
  • Rules for probability distribution:
    • 0 <= p(x) <= 1
    • ∑ p(x) = 1
expected value
Expected Value
  • Expected value (or population mean) of a random variable x with the probability distribution p(x) is

Intuition:expected value is weighted average of x

variance of a random variable
Variance of a Random Variable
  • The variance of random variable x with probability distribution p(x) and expected value E(x)= is given as
  • The Standard Deviation of random variable x is equal to the square root of its variance.