Binomial Distribution - Business Statistics - Lecture Slides, Slides of Business Statistics

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The Binomial Distribution

October 20, 2010

Bernoulli Trials

Definition A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure.

(^1) Tossing a coin and considering heads as success and tails as failure.

Bernoulli Trials

Definition A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure.

(^1) Tossing a coin and considering heads as success and tails as failure. (^2) Checking items from a production line: success = not defective, failure = defective. (^3) Phoning a call centre: success = operator free; failure = no operator free.

Bernoulli Random Variables

A Bernoulli random variable X takes the values 0 and 1 and

P(X = 1) = p P(X = 0) = 1 − p.

It can be easily checked that the mean and variance of a Bernoulli random variable are

E (X ) = p V (X ) = p(1 − p).

Binomial Experiments

Consider the following type of random experiment: (^1) The experiment consists of n repeated Bernoulli trials - each trial has only two possible outcomes labelled as success and failure; (^2) The trials are independent - the outcome of any trial has no effect on the probability of the others;

Binomial Experiments

Consider the following type of random experiment: (^1) The experiment consists of n repeated Bernoulli trials - each trial has only two possible outcomes labelled as success and failure; (^2) The trials are independent - the outcome of any trial has no effect on the probability of the others; (^3) The probability of success in each trial is constant which we denote by p.

The Binomial Distribution

Definition The random variable X that counts the number of successes, k, in the n trials is said to have a binomial distribution with parameters n and p, written bin(k; n, p).

The probability mass function of a binomial random variable X with parameters n and p is

f (k) = P(X = k) =

n k

pk^ (1 − p)n−k

for k = 0, 1 , 2 , 3 ,... , n.

The Binomial Distribution

Definition The random variable X that counts the number of successes, k, in the n trials is said to have a binomial distribution with parameters n and p, written bin(k; n, p).

The probability mass function of a binomial random variable X with parameters n and p is

f (k) = P(X = k) =

n k

pk^ (1 − p)n−k

for k = 0, 1 , 2 , 3 ,... , n. (n k

counts the number of outcomes that include exactly k successes and n − k failures.

Binomial Distribution - Examples

Example (i) If we call heads a success then this X has a binomial distribution with parameters n = 6 and p = 0.3.

P(X = 2) =

(0.3)^2 (0.7)^4 = 0. 324135

Binomial Distribution - Examples

Example (i) If we call heads a success then this X has a binomial distribution with parameters n = 6 and p = 0.3.

P(X = 2) =

(0.3)^2 (0.7)^4 = 0. 324135

(ii)

P(X = 3) =

(0.3)^3 (0.7)^3 = 0. 18522.

(iii) We need P(1 < X ≤ 5)

Binomial Distribution - Example

Example A quality control engineer is in charge of testing whether or not 90% of the DVD players produced by his company conform to specifications. To do this, the engineer randomly selects a batch of 12 DVD players from each day’s production. The day’s production is acceptable provided no more than 1 DVD player fails to meet specifications. Otherwise, the entire day’s production has to be tested. (i) What is the probability that the engineer incorrectly passes a day’s production as acceptable if only 80% of the day’s DVD players actually conform to specification? (ii) What is the probability that the engineer unnecessarily requires the entire day’s production to be tested if in fact 90% of the DVD players conform to specifications?

Example

Example (i) Let X denote the number of DVD players in the sample that fail to meet specifications. In part (i) we want P(X ≤ 1) with binomial parameters n = 12, p = 0.2.

P(X ≤ 1) = P(X = 0) + P(X = 1)

=

(0.2)^0 (0.8)^12 +

(0.2)^1 (0.8)^11

Binomial Distribution - Mean and Variance

(^1) Any random variable with a binomial distribution X with parameters n and p is a sum of n independent Bernoulli random variables in which the probability of success is p.

X = X 1 + X 2 + · · · + Xn.

Binomial Distribution - Mean and Variance

(^1) Any random variable with a binomial distribution X with parameters n and p is a sum of n independent Bernoulli random variables in which the probability of success is p.

X = X 1 + X 2 + · · · + Xn.

(^2) The mean and variance of each Xi can easily be calculated as:

E (Xi ) = p, V (Xi ) = p(1 − p).