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Discrete Probability



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Chapter 7 : Discrete Probability  



Discrete Probability arrow_upward


  • The Probability Distribution for a random variable describes how probabilities are distributed over the values of the random variable.
    • We can describe a Discrete Probability Distribution with a table, graph, or equation.
  • Discrete outcomes are:
    • Mutually exclusive (nothing in common).
    • Collectively exhaustive (nothing left out).
  • Discrete Probability  ranges from 0 to 1.
  •    

       

  • Discrete Probability Distribution includes the following:

  • Random Variables arrow_upward


  • A Random Variable is a numerical description of the outcome of an experiment.

  • Types of Random Variable:

    Discrete Random Variable:
  • A Discrete Random Variable may assume either a finite number of values or an infinite sequence of values.
  • Continuous Random Variable:
  • A Continuous Random Variable may assume any numerical value in an interval or collection of intervals.

  • Expected Value (The Mean) arrow_upward


  • It is a weighted average of the Probability Distribution.
  • Multiplying the P of each outcome by the value of the outcome, and then summing the results.
  •    

       

    Example:
  • Toss 2 coins, count the number of tails and compute expected value.
  • Solution:

  • Values

    Probability

    0

    0.25

    1

    0.50

    2

    0.25


        

        


    Variance arrow_upward


  • The weighted average squared deviation about the mean.
  •     

       

    Example:
  • Toss 2 coins, count the number of tails and compute variance.
  • Solution:

    Values

    Probability

    0

    0.25

    1

    0.50

    2

    0.25



    Standard Deviation arrow_upward


  • The Standard Deviation is the square root of the Variance.

  • Co-variance arrow_upward


  • Co-variance is a statistic representation of the degree to which two variables vary together.
  • It shows the type and strength of the relationship between two variables.
  • Positive
    • Indicates that the two variables will vary together in the same direction.
  • Negative
    • Indicates that the two variables will vary in opposite directions.

    Where,

     Discrete Random Variable.

     Outcome of .

    Discrete Random Variable.

     Outcome of .

     Probability of occurrence of the  outcome of and the  outcome of .


    Binomial Probability Distribution arrow_upward


  • It is used to find the probability that a given number of successes X occurs over a certain number of trialsn, each of which has the same probability p.
  • Success refers to the event of interest to us, such as:
    • Getting a head when flipping a coin.
    • Getting a question correct on a quiz.

    Binomial Distribution: Formula arrow_upward


  • Let x denote the number of successes occurring in the n trials.
  • Where,
    • f(x) = The probability of x successes in n trials.
    • p = The probability of success on any one trial.

    Mean of Binomial Distribution arrow_upward


  • Let X be a discrete random variable with the binomial distribution with parameters n and p.
  • The mean of binomial Distribution is given by:
  •    

     


    Variance and Standard Deviation of Binomial Distribution arrow_upward


  • Let X be a discrete random variable with the binomial distribution with parameters n and p.
  • The variance and standard deviation of binomial Distribution is given by:

  • Poisson Distribution arrow_upward


  • The Poisson Distribution is used to model the number of events occurring within a given time interval.

  • Properties of Poisson Experiment:

  • The probability of an occurrence is the same for any two intervals of equal length.
  • The expected value of occurrences in an interval is proportional to the length of this interval.
  • The occurrence or non-occurrence in any interval is independent of the occurrence or non-occurrence in any other interval.
  • The probability of two or more occurrences in a very small interval is close to 0.

  • Poisson Distribution: Formula arrow_upward


    Where,

     is an average rate of value.
    x is a poisson random variable.
    e is the base of logarithm (e = 2.718).


    Hyper-Geometric Distribution arrow_upward


  • The Hyper-Geometric Distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement.
  • Where,
    • N = Population size.
    • A = Number of successes in the population.
    • N – A = Number of failures in the population.
    • n = Sample size.
    • X = Number of successes in the sample.
    • n – X = Number of failures in the sample.


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