Diese Präsentation wurde erfolgreich gemeldet.
Wir verwenden Ihre LinkedIn Profilangaben und Informationen zu Ihren Aktivitäten, um Anzeigen zu personalisieren und Ihnen relevantere Inhalte anzuzeigen. Sie können Ihre Anzeigeneinstellungen jederzeit ändern.

Bernoullis Random Variables And Binomial Distribution

3.812 Aufrufe

Veröffentlicht am

Bernoullis Random Variables And Binomial Distribution

Veröffentlicht in: Bildung
  • Als Erste(r) kommentieren

Bernoullis Random Variables And Binomial Distribution

  1. 1. 1.10 Bernoulli’s random Variables & Binomial Distribution<br />
  2. 2. Bernoulli Random Variable<br />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<br />
  3. 3. Bernoulli Random Variable<br /> 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 <br />
  4. 4. Binomial Random Variable<br />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) .<br />
  5. 5. Binomial Distribution<br />Bernoulli Trials<br />There are only two possible outcomes for each trial.<br />The probability of a success is the same for each trial.<br />There are n trials, where n is a constant.<br />The n trials are independent.<br />
  6. 6. Binomial Distribution <br />Let X be the random variable that equals the number of successes in n trials.<br />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 <br />px(1- p)n – x<br />The number of ways in which one can select the x trials on which there is to be a success is <br />
  7. 7. Binomial Distribution <br />Thus the probability of getting x successes in n trials is given by<br />This probability distribution is called the binomial <br />distribution because for x = 0, 1, 2, …, and n the <br />value of the probabilities are successive terms of <br />binomial expansion of [p + (1 – p)]n;<br />
  8. 8. Binomial Distribution <br />for the same reason, the combinatorial quantities <br />are referred to as binomial coefficients. <br />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.<br />
  9. 9. Binomial Distribution <br />Distribution function for binomial distribution<br />
  10. 10. Binomial Distribution <br />The value of b(x;n,p) can be obtained by formula<br />since the two cumulative probabilities B(x; n, p) and B(x - 1; n, p) differ by the single term b(x; n,p).<br />If n is large the calculation of binomial probability can become quite tedious.<br />
  11. 11. Binomial Distribution Function<br />Table for n = 2 and 3 and p = .05 to .25<br />
  12. 12. Example<br />
  13. 13. The Mean and the Variance of a Probability Distribution<br />Mean of discrete probability distribution<br />The mean of a probability distribution is the mathematical expectation of a corresponding random variable. <br />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 <br /> = x1· f(x1) + x2· f(x2) + … + xk· f(xk) <br />
  14. 14. The Mean and the Variance of a Probability Distribution <br />Mean of binomial distribution<br />p  probability of success<br />n  number of trials<br />Variance of binomial distribution<br />
  15. 15. The Mean and the Variance of a Probability Distribution <br />Mean of binomial distribution<br />p  probability of success<br />n  number of trials<br />Proof:<br />
  16. 16. The Mean and the Variance of a Probability Distribution <br />Put x – 1= y and n – 1 = m, so n – x = m – y, <br />
  17. 17. Computing formula for the variance<br />Variance of binomial distribution<br />Proof:<br />
  18. 18. Put x – 1 = y and n – 1 = m <br />The Mean and the Variance of a Probability Distribution <br />
  19. 19. The Mean and the Variance of a Probability Distribution <br />
  20. 20. Put y – 1 = z and m – 1 = l in first summation<br />The Mean and the Variance of a Probability Distribution <br />
  21. 21. Moment Generating function for Binomial distribution<br />
  22. 22. Second ordinary/raw moment (moment about origin)<br />Moment Generating function for Binomial distribution<br />
  23. 23. Moment Generating function for Binomial distribution<br />Moment Generating function for Binomial distribution<br />

×