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Lecture 07 - Binomial Distributions (5.2-5.3)

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Binomial Distributions

How to recognize binomial random variable:

1. The sample size n is fixed.
2. The n observations (or trials) are independent.
3. There are only 2 possible outcomes for each observation. They are labeled “Success” and “Failure”
4. The probability of success is the same for each trial. Let p = success probability and 1 – p = failure probability
5. The binomial random variable is . . . X = the number of successes out of n observations.

Binomial distributions are identified specifically by two parameters: n and p.
X takes on values 0, 1, 2, . . . , n

Notation: [pic]

Mean: [pic]

Variance: [pic]

Standard deviation: [pic]

The Binomial Probability Formula:

[pic]

Suppose [pic], then the probability of observing 16 successes out of 20 observations is
[pic]

Find the mean, variance, and standard deviation of X.

Mean: [pic]

Variance: [pic]

Standard deviation: [pic]

Table 3 in Appendix II contains probability distributions for a variety of binomial distributions.

Look up the following probabilities: For X ~ B(20, 0.7), find P(X=16). Answer: P(X=16) = 0.130

For X ~ B(8, 0.45), find P(X > 5). Answer: P(X > 5) = P(6) + P(7) + P(8) = 0.070 + 0.016 + 0.002 = 0.088
More examples:
1. A research team found that 10% of all people who wear ties wear them so tight that they actually reduce blood flow to the brain. In a random sample of 9 people wearing ties, what is the probability that . . . a. no ties are too tight? [pic]
[pic]

b. at least one tie is too tight? [pic]

c. more than two ties are too tight?
[pic]
[pic]

2. If X ~ B(12, 0.37) find P(x = 5).
[pic]
The binomial “quota” problem:

Exercise 20. A large bank vault has several automatic burglar alarms. The probability is 0.55 that a single alarm will detect a burglar. (assume the alarms work independently)

How many alarms should be used to be 99% certain that a burglar is detected by at least one alarm?

Solution: Let X ~ B(n, 0.55) and use Table 3 to see for which value of n is
[pic]

[pic]
[pic]
[pic]
[pic]
[pic]
Finally! We see that n = 6 alarms will do it.

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