(Keller 7, 8, 9)

### Probability distributions

(Keller 7, 8)

A probability distribution is a table, formula or graph that describes the values of a random variable and the probability associated with these values. Since a random variable can be either discrete or continuous we have two types of probability distributions: discrete and continuous probability distributions.

**Example**

The following formula represents the continuous normal distribution function.

**Example**

A distribution function can also be described by a table as is the case with the number of persons in a US household..

Number of persons | Number of households |
---|---|

1 | 31.1 |

2 | 38.6 |

3 | 18.8 |

4 | 16.2 |

5 | 7.2 |

6 | 2.7 |

7 or more | 1.4 |

Total | 116.0 |

### Discrete probability distributions

(Keller 7)

The following requirements hold for the probability :

- for all ;
- .

There is a relation between a relative frequency diagram and a discrete probability function.

**Example**

The probability distribution can be estimated from relative frequencies.

is a discrete variable, the number of persons in a household.

X | # households millions | P(x) |

1 | 31.1 | .268 |

2 | 38.6 | .333 |

3 | 18.8 | .162 |

4 | 16.2 | .140 |

5 | 7.2 | .062 |

6 | 2.7 | .023 |

7 or more | 1.4 | .012 |

Total | 116.0 | 1.00 |

is the discrete probability distribution of the number of persons in a household.

We have: , , etc.

Also we can compute

### Population mean *E(X)*

(Keller 7)

The population mean is the weighted average of all values of . The weights are the probabilities.

is called the expected value of and is defined by the following formula.

**Example**

What is the mean of throws of a fair dice?

for all , because it is a fair dice.

Applying the formula we get:

### Population variance *V(X)*

(Keller 7)

The population variance is calculated similarly. It is the weighted average of the squared deviations from the mean . The weights are the probabilities. It is defined by the following formula:

**Example**

We compute the variance of the households example. First we compute:

Now we can compute the variance:

The standard deviation is .

### Covariance of two discrete variables

The covariance of two discrete variables and is defined as:

is the joint probability distribution of the random variables and : **and **.

Note. We also write .

### Laws of *E(X*) and *V(X)*

(Keller 7)

The following formulas can easily be derived from the definitions of and .

For example:

### Laws about a linear sum

(Keller, p. 234, and are two random variables).

If and are independent then and thus:

**Example**

If and are independent then, because and :

### Coefficient of correlation

The coefficient of correlation between two variables and is defined as the covariance divided by the standard deviations of the variables.

The population coefficient of correlation is:

The sample coefficient of correlation is:

The coefficient of correlation answers the question: how strong is the association between and ?

The advantage of the coefficient of correlation over the covariance is that it has a fixed range from to (proven by Mathematics). If the two variables are very strongly and positively related, the coefficient value is close to (strong positive linear relationship). If the two variables are very strongly and negatively related, the coefficient value is close to (strong negative linear relationship). No straight linear relationship is indicated by a coefficient close to 0.

The following graphs depict the relations of and for various coefficients of correlation, varying from to . Below a number of examples.

### Binomial distribution

(Keller 7)

The binomial distribution is the probability distribution that results from doing a binomial experiment. Binomial experiments have the following properties:

- There are a fixed number of trials, represented as ;
- Each trial has
**two**possible outcomes, success or failure; - (success); (failure
- The trials are independent, meaning that the outcome of one trial does not affect the outcomes of any other trials.

The binomial random variable counts the number of successes in trials of the binomial experiment.

(e.g. s s f s f f f s f s s f s* *shows trials and successes).

To calculate the probability associated with each value we use combinatorics:

for

**Example**

A quiz consists of independent multiple-choice questions (). Each question has possible answers, only one of which is correct (). You choose to guess the answer to each question. is the number of correct guesses . The probability that you will have a score is:

{ is called factorial ; ; ).

The mean, variance and standard deviation of a binomial random variable are (derived mathematically):

and thus:

### Continuous random variables

(Keller 8)

Unlike a discrete random variable, a continuous random variable is one that assumes an uncountable number of values. We cannot list the possible values because there is an infinite number of them. Because there is an infinite number of values, the probability of each individual value is . The probability that a man has a height of exactly 180 cm is:

### Pobability density functions

{Keller 8)

A function is called a probability density function (over the range ) if it meets the following requirements:

for all

and the total area between curve and -axis is:

For the interval [a, b] we may also take , as is the case in e.g. the normal distribution.

### The normal density function

(Keller 8)

The normal distribution is the most important of all probability distributions. The probability density function of a normal random variable is given by:

for

The graph is bell-shaped and symmetrical around the mean . This density function is also denoted by or .

The normal distribution function is defined by:

Therefore, the probability equals .

This infinite integral cannot be computed analytically (pen and paper), therefore we need a table or a computer can do the job.

### Standard normal distribution

(Keller 8)

A normal density function with mean and standard deviation is called the standard normal density.

for

Any normal distribution can be converted to a standard normal distribution, see below. The standard normal distribution is also denoted by . Any (normal) variable can be converted to a new (normal) variable :

with the following properties:

.

Thus, if

then

.

**Example**

Suppose the demand is a normally distributed variable with mean and standard deviation and we want to compute . Then:

.

The answer can be found in Table 3 of Appendix B9 of Keller, or by Excel.

### Other continuous distributions

There are three other continuous distributions which will be used and explained later.

- distribution (also called Student's distribution);
- (Chi-squared) distribution;
- distribution.