Binomial Distribution
The Binomial Distribution
October 20, 2010
The Binomial Distribution
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.
The Binomial Distribution
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. Checking items from a production line: success = not defective, failure = defective.
2
The Binomial Distribution
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. Checking items from a production line: success = not defective, failure = defective. Phoning a call centre: success = operator free; failure = no operator free.
2
3
The Binomial Distribution
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).
The Binomial Distribution
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;
The Binomial Distribution
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; The trials are independent - the outcome of any trial has no effect on the probability of the others;
2
The Binomial Distribution
Binomial Experiments
Consider the following
October 20, 2010
The Binomial Distribution
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.
The Binomial Distribution
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. Checking items from a production line: success = not defective, failure = defective.
2
The Binomial Distribution
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. Checking items from a production line: success = not defective, failure = defective. Phoning a call centre: success = operator free; failure = no operator free.
2
3
The Binomial Distribution
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).
The Binomial Distribution
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;
The Binomial Distribution
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; The trials are independent - the outcome of any trial has no effect on the probability of the others;
2
The Binomial Distribution
Binomial Experiments
Consider the following