1. THE BINOMIAL RANDOM VARIABLE

2. BERNOULLI RANDOM VARIABLE BernoulliA random variable X with the following properties is called a Bernoulli random variable: –P(X = 1) = p;P(X = 0) = 1-p p = P(success or good or bad or yes or no or 1, etc.) Mean and Variance of a Bernoulli random variable, X E(X)p E(X) = 1p +0(1-p) = p Var(X)p(1-p) Var(X) = (1 2 p + 0 2 (1-p)) - p 2 = p- p 2 = p(1-p)

3. BINOMIAL RANDOM VARIABLE nindependently p binomialWhen n items are sampled independently, each of which has a probability p of being a success, the number of successes is said to be a binomial random variable X = number of successes in n tries Thus, X = X 1 + X 2 + … + X n, where Bernoulli X 1, X 2, … X n are all Bernoulli random variables with means of p and variances of p(1-p)

4. BINOMIAL MEAN AND VARIANCE E(X)E(X) = E(X 1 +X 2 +…+X n ) = np E(X 1 ) + E(X 2 ) + … E(X n ) = p+p+…+p = np Since X 1, X 2 …X n are independent random variables Var(X) Var(X) = Var(X 1 +X 2 +…+X n ) = np(1-p) Var(X 1 ) + Var(X 2 ) + … + Var(X n ) = p(1-p) + p(1-p) + … + p(1-p) = np(1-p)

5. EXAMPLE OF A BINOMIAL RANDOM VARIABLE Distribution of the number of bad batteries in a sample of 4 where each battery has a chance of.1 of being bad (B) (and.9 of being good(G)) X = # bad batteries in a sample of 4

6. BINOMIAL DISTRIBUTION XWays of getting X Prob(Each Way) Prob 0GGGG (.9) 4 =.6561 1BGGG, GBGG, (.1)(.9) 3 =.0729 GGBG, GGGB 4(.0729) =.2916 2BBGG, BGBG, (.1) 2 (.9) 2 =.0081 BGGB, GBGB GBBG, GGBB 6(.0081) =.0486 3BBBG, BBGB, (.1) 3 (.9) =.0009 BGBB, GBBB 4(.0009) =.0036 4 BBBB (.1) 4 =.0001

7. BINOMIAL DISTRIBUTION GENERAL CASE X = of successes in n tries when the probability of success on any try is p P(X = x)P(X = x) = p x (1-p) n-x (# ways of getting x successes in n tries) p x (1-p) n-x Note: In the previous example a “success” was getting a “bad” battery

8. # Ways of Getting x Successes in n Tries n! = product of all positive integers that are  n, e.g. 5! = 5(4)(3)(2)(1) = 120 Note: by definition 0! = 1

9. EXAMPLE What is the probability of getting exactly 2 bad batteries in a random sample of size 15 when the likelihood that any battery is bad is.1?

10. Calculating a Binomial Probability from the Binomial Formula

11. Example of a Cumulative Binomial Probability What is the probability of getting at most 2 bad batteries in a random sample of 15 when the probability that any battery is bad is.1?

12. The Calculation of a Cumulative Probability

13. Example (Cont’d) What is the mean number of bad batteries in a random sample of 15? –  1.5 –  = np = 15(.1) = 1.5 What is the standard deviation of the number of bad batteries in a sample of 15? –Var(X)1.35 –Var(X) = np(1-p) = 15(.1)(.9) = 1.35 –Standard Deviation1.162 –Standard Deviation = SQRT(1.35) = 1.162

14. BINOMIAL PROBABILITIES USING EXCEL Sample size = n; probability of success = p P(EXACTLY x successes) = BINOMDIST(x,n,p,FALSE) P(x OR LESS successes) =BINOMDIST(x,n,p,TRUE)

15. Point and Cumulative Probabilities When n = 15, p =.1 =BINOMDIST(3,15,.1,FALSE) =BINOMDIST(5,15,.1,TRUE)

16. Typical Binomial Probabilities “Less Than or Equal”Prob. =BINOMDIST(2,15,.1,True) “Between” Prob. =BINOMDIST(4,15,.1,TRUE)-BINOMDIST(1,15,.1,TRUE) “Greater Than or Equal to” Prob =1–BINOMDIST(3,15,.1,TRUE) “Equal To” Prob. “Equal To” Prob.=BINOMDIST(3,15,.1,FALSE)

17. REVIEW Bernoulli random variable –Definition, Mean, Variance Binomial Random Variable –Definition –Mean, Variance, Standard Deviation –Point and Cumulative Probabilities Using the Binomial Formula Using Excel