Additional Properties of the Binomial Distribution

Graphing a Binomial Distribution

The table shows a binomial experiment with trials, and

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2 0.060

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5 0.303

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Additional Properties of the Binomial Distribution

Graphing a Binomial Distribution

The table shows a binomial experiment with trials, and

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2 0.060

3 0.185

4 0.324

5 0.303

6 0.118

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Additional Properties of the Binomial Distribution

Graphing a Binomial Distribution

The table shows a binomial experiment with trials, and

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5 0.303

6 0.118

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Additional Properties of the Binomial Distribution

Graphing a Binomial Distribution

The table shows a binomial experiment with trials, and

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Additional Properties of the Binomial Distribution

Graphing a Binomial Distribution

The table shows a binomial experiment with trials, and

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Additional Properties of the Binomial Distribution

Graphing a Binomial Distribution

The table shows a binomial experiment with trials, and

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Additional Properties of the Binomial Distribution

Graphing a Binomial Distribution

The table shows a binomial experiment with trials, and

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2 0.060

3 0.185

4 0.324

5 0.303

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Additional Properties of the Binomial Distribution

Graphing a Binomial Distribution

The table shows a binomial experiment with trials, and

0 0.001

1 0.010

2 0.060

3 0.185

4 0.324

5 0.303

6 0.118

1. Place values on the x – axis

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the corresponding height of the The bar will have its middle over the

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Additional Properties of the Binomial Distribution

Graphing a Binomial Distribution

The table shows a binomial experiment with trials, and

0 0.001

1 0.010

2 0.060

3 0.185

4 0.324

5 0.303

6 0.118

1. Place values on the x – axis

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the corresponding height of the The bar will have its middle over the

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Additional Properties of the Binomial Distribution

Graphing a Binomial Distribution

The table shows a binomial experiment with trials, and

0 0.001

1 0.010

2 0.060

3 0.185

4 0.324

5 0.303

6 0.118

1. Place values on the x – axis

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the corresponding height of the The bar will have its middle over the

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Additional Properties of the Binomial Distribution

Graphing a Binomial Distribution

The table shows a binomial experiment with trials, and

0 0.001

1 0.010

2 0.060

3 0.185

4 0.324

5 0.303

6 0.118

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the corresponding height of the The bar will have its middle over the

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Additional Properties of the Binomial Distribution

How to compute and for a binomial distribution :

- the expected number of successes for random variable

Additional Properties of the Binomial Distribution

How to compute and for a binomial distribution :

- the expected number of successes for random variable

- the standard deviation for random variable

Additional Properties of the Binomial Distribution

How to compute and for a binomial distribution :

- the expected number of successes for random variable

- the standard deviation for random variable

Also : - is a random variable representing the number of successes- is the number of trials- is the probability of success on a single trial- and is the probability of failure on a single trial

Additional Properties of the Binomial Distribution

How to compute and for a binomial distribution :

- the expected number of successes for random variable

- the standard deviation for random variable

Also : - is a random variable representing the number of successes- is the number of trials- is the probability of success on a single trial- and is the probability of failure on a single trial

EXAMPLE : Find the mean and standard deviation given :

Additional Properties of the Binomial Distribution

How to compute and for a binomial distribution :

- the expected number of successes for random variable

- the standard deviation for random variable

Also : - is a random variable representing the number of successes- is the number of trials- is the probability of success on a single trial- and is the probability of failure on a single trial

EXAMPLE : Find the mean and standard deviation given :

Solution :

Additional Properties of the Binomial Distribution

How to compute and for a binomial distribution :

- the expected number of successes for random variable

- the standard deviation for random variable

Also : - is a random variable representing the number of successes- is the number of trials- is the probability of success on a single trial- and is the probability of failure on a single trial

EXAMPLE : Find the mean and standard deviation given :

Solution :

Additional Properties of the Binomial Distribution

Chebyshev’s theorem tells us that 75% of all data falls within 2 standard deviations of the mean. As we will see later, actually 95% of all data will fall within 2 standard deviations of the mean. So if the mean = 12, and the standard deviation = 2, 95% of all data will fall in between 8 and 16.

Additional Properties of the Binomial Distribution

Chebyshev’s theorem tells us that 75% of all data falls within 2 standard deviations of the mean. As we will see later, actually 95% of all data will fall within 2 standard deviations of the mean. So if the mean = 12, and the standard deviation = 2, 95% of all data will fall in between 8 and 16.

A data item outside 2 standard deviations is called an outlier. It is less common than the rest of the data. We will look at these scenarios in a later chapter.

Additional Properties of the Binomial Distribution

BINOMIAL PROBABILITIES - EQUIVALENT FORMS

We need a method for expressing binomial probabilities for multiple outcomes. For example, if we want to find , we could use

Additional Properties of the Binomial Distribution

BINOMIAL PROBABILITIES - EQUIVALENT FORMS

We need a method for expressing binomial probabilities for multiple outcomes. For example, if we want to find , we could use

How about ?

Additional Properties of the Binomial Distribution

BINOMIAL PROBABILITIES - EQUIVALENT FORMS

We need a method for expressing binomial probabilities for multiple outcomes. For example, if we want to find , we could use

How about ?

Additional Properties of the Binomial Distribution

BINOMIAL PROBABILITIES - EQUIVALENT FORMS

We need a method for expressing binomial probabilities for multiple outcomes. For example, if we want to find , we could use

How about ?

Can you find

Additional Properties of the Binomial Distribution

BINOMIAL PROBABILITIES - EQUIVALENT FORMS

We need a method for expressing binomial probabilities for multiple outcomes. For example, if we want to find , we could use

How about ?

Can you find

Additional Properties of the Binomial Distribution

BINOMIAL PROBABILITIES - EQUIVALENT FORMS

We need a method for expressing binomial probabilities for multiple outcomes. For example, if we want to find , we could use

How about ?

Can you find

Find

Additional Properties of the Binomial Distribution

BINOMIAL PROBABILITIES - EQUIVALENT FORMS

We need a method for expressing binomial probabilities for multiple outcomes. For example, if we want to find , we could use

How about ?

Can you find

Find

Additional Properties of the Binomial Distribution

BINOMIAL PROBABILITIES - EQUIVALENT FORMS

We need a method for expressing binomial probabilities for multiple outcomes. For example, if we want to find , we could use

How about ?

Can you find

Find

IN GENERAL :

Additional Properties of the Binomial Distribution

BINOMIAL PROBABILITIES - EQUIVALENT FORMS

We need a method for expressing binomial probabilities for multiple outcomes. For example, if we want to find , we could use

How about ?

Can you find

Find

IN GENERAL :

EXAMPLE : Find if

Additional Properties of the Binomial Distribution

BINOMIAL PROBABILITIES - EQUIVALENT FORMS

We need a method for expressing binomial probabilities for multiple outcomes. For example, if we want to find , we could use

How about ?

Can you find

Find

IN GENERAL :

EXAMPLE : Find if

Using a binomial table where

Additional Properties of the Binomial Distribution

BINOMIAL PROBABILITIES - EQUIVALENT FORMS

We need a method for expressing binomial probabilities for multiple outcomes. For example, if we want to find , we could use

How about ?

Can you find

Find

IN GENERAL :

EXAMPLE : Find if

Using a binomial table where

Additional Properties of the Binomial Distribution

BINOMIAL PROBABILITIES - EQUIVALENT FORMS

We need a method for expressing binomial probabilities for multiple outcomes. For example, if we want to find , we could use

How about ?

Can you find

Find

IN GENERAL :

EXAMPLE : Find if

Using a binomial table where