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.

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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.

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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:

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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;

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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; The probability of success in each trial is constant which we denote by p.

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The Binomial Distribution

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

The Binomial Distribution

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) = for k = 0, 1, 2, 3, . . . , n. n k p (1 − p)n−k k

The Binomial Distribution

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) = for k = 0, 1, 2, 3, . . . , n. counts the number of outcomes that include exactly k successes and n − k failures. n k

n k p (1 − p)n−k k

The Binomial Distribution

Binomial Distribution - Examples

Example A biased coin is tossed 6 times. The probability of heads on any toss is 0.3. Let X denote the number of heads that come up. Calculate: (i) P(X = 2) (ii) P(X = 3) (iii) P(1 < X ≤ 5).

The Binomial Distribution

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)...