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Bernoullis Random Variables And Binomial Distribution
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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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    Bernoullis Random Variables And Binomial Distribution Bernoullis Random Variables And Binomial Distribution Presentation Transcript

    • 1.10 Bernoulli’s random Variables & Binomial Distribution
    • 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
    • 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
    • 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) .
    • 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.
    • 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
    • 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;
    • 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.
    • Binomial Distribution
      Distribution function for binomial distribution
    • 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.
    • Binomial Distribution Function
      Table for n = 2 and 3 and p = .05 to .25
    • Example
    • 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)
    • 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
    • The Mean and the Variance of a Probability Distribution
      Mean of binomial distribution
      p  probability of success
      n  number of trials
      Proof:
    • The Mean and the Variance of a Probability Distribution
      Put x – 1= y and n – 1 = m, so n – x = m – y,
    • Computing formula for the variance
      Variance of binomial distribution
      Proof:
    • Put x – 1 = y and n – 1 = m
      The Mean and the Variance of a Probability Distribution
    • The Mean and the Variance of a Probability Distribution
    • Put y – 1 = z and m – 1 = l in first summation
      The Mean and the Variance of a Probability Distribution
    • Moment Generating function for Binomial distribution
    • Second ordinary/raw moment (moment about origin)
      Moment Generating function for Binomial distribution
    • Moment Generating function for Binomial distribution
      Moment Generating function for Binomial distribution