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Bernoullis Random Variables And Binomial Distribution

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Bernoullis Random Variables And Binomial Distribution

Bernoullis Random Variables And Binomial Distribution

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  • 1. 1.10 Bernoulli’s random Variables & Binomial Distribution
  • 2. Bernoulli Random Variable
    Suppose that a trial, or an experiment, whose outcome can be classified as either a success or a failure is performed. If we let X=1 when the outcome is a success and X=0 when the outcome is a failure, then the pmf of X is given by
  • 3. Bernoulli Random Variable
    A random variable X is said to be a Bernoulli random variable (after the Swiss mathematician James Bernoulli) if its probability mass function is given by
  • 4. Binomial Random Variable
    Suppose now that n independent trials, each of which results in a success with probability p and in a failure with probability 1-p, are to be performed. If X represents the number of successes that occur in the n trials, then X is said to be a Binomial random variable with parameters (n,p) . Thus a Bernoulli random variable is just a binomial random variable with parameters (1,p) .
  • 5. Binomial Distribution
    Bernoulli Trials
    There are only two possible outcomes for each trial.
    The probability of a success is the same for each trial.
    There are n trials, where n is a constant.
    The n trials are independent.
  • 6. Binomial Distribution
    Let X be the random variable that equals the number of successes in n trials.
    If p and 1 – p are the probabilities of success and failure on any one trial then the probability of getting x successes and n – x failures in some specific order is
    px(1- p)n – x
    The number of ways in which one can select the x trials on which there is to be a success is
  • 7. Binomial Distribution
    Thus the probability of getting x successes in n trials is given by
    This probability distribution is called the binomial
    distribution because for x = 0, 1, 2, …, and n the
    value of the probabilities are successive terms of
    binomial expansion of [p + (1 – p)]n;
  • 8. Binomial Distribution
    for the same reason, the combinatorial quantities
    are referred to as binomial coefficients.
    The preceding equation defines a family of probability distributions with each member characterized by a given value of the parameterp and the number of trials n.
  • 9. Binomial Distribution
    Distribution function for binomial distribution
  • 10. Binomial Distribution
    The value of b(x;n,p) can be obtained by formula
    since the two cumulative probabilities B(x; n, p) and B(x - 1; n, p) differ by the single term b(x; n,p).
    If n is large the calculation of binomial probability can become quite tedious.
  • 11. Binomial Distribution Function
    Table for n = 2 and 3 and p = .05 to .25
  • 12. Example
  • 13. The Mean and the Variance of a Probability Distribution
    Mean of discrete probability distribution
    The mean of a probability distribution is the mathematical expectation of a corresponding random variable.
    If a random variable X takes on the values x1, x2, …, or xk, with the probability f(x1), f(x2),…, and f(xk), its mathematical expectation or expected value is
     = x1· f(x1) + x2· f(x2) + … + xk· f(xk)
  • 14. The Mean and the Variance of a Probability Distribution
    Mean of binomial distribution
    p  probability of success
    n  number of trials
    Variance of binomial distribution
  • 15. The Mean and the Variance of a Probability Distribution
    Mean of binomial distribution
    p  probability of success
    n  number of trials
    Proof:
  • 16. The Mean and the Variance of a Probability Distribution
    Put x – 1= y and n – 1 = m, so n – x = m – y,
  • 17. Computing formula for the variance
    Variance of binomial distribution
    Proof:
  • 18. Put x – 1 = y and n – 1 = m
    The Mean and the Variance of a Probability Distribution
  • 19. The Mean and the Variance of a Probability Distribution
  • 20. Put y – 1 = z and m – 1 = l in first summation
    The Mean and the Variance of a Probability Distribution
  • 21. Moment Generating function for Binomial distribution
  • 22. Second ordinary/raw moment (moment about origin)
    Moment Generating function for Binomial distribution
  • 23. Moment Generating function for Binomial distribution
    Moment Generating function for Binomial distribution

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