Download 1 # 2 K 3 Binomial distribution

Distributions
$ # K
Binomial distribution
A binomial distribution is defined by two parameters, n and p.
The number of trials is n. An experiment consist of n trials. Each trial can only have two outcomes:
success or failure
The probability of success is p.
The distribution, the expected value and the standard deviation depend only on the parameters n and p.
P(X=k) is the probability of k successes.
The probability distribution of the random variable X is shown in the form of a bar chart and in the form
of a table. In the table is also given the cumulative probability P(X<=k) and the probability P(X>=k). By
clicking in a cell of the table, you see the corresponding part in the graph in red.
There are two options available:
Cumulative
The bar chart shows the cumulative distribution. However, the color connection between table and table
is not now active.
Normal approximation
The best fit normal, or cumulative normal, distribution is superimposed on the graph of the binomial
distribution.
$
Binomiale verdeling
Binomialeverdeling
K Binomial distribution
#
$ # K
Normal distribution
A normal distribution is defined by two parameters, μ (mu) the mean and σ (sigma) the standard
deviation.
The probability distribution is a continuous, symmetric, uni-modal distribution, where only probabilities
like P(X<=b) or P(X>=b) make sense
The normal distribution with μ = 0 and σ = 1 is called the standard normal distribution. This distribution is
the starting point on the screen, but you can give ì any value, and ó any positive value, and the x-axis will
be adjusted.
You can drag the bounds between the blue and yellow areas.
You can change the value of μ by typing or by dragging the red circle.
You can change the value of σ by typing or by dragging the green circle.
The principle is P(X<=bound | μ; σ ) = probability corresponding to the blue area.
There is a connection between the four parameters: bound(s), probability, μ and σ . You can see the
connection if you change/drag one of these items.
Always one of these values is unknown. That is the value that is looked for and is determined by the
values that are given to the other parameters.
There are a number of options:
Adapt graph
This is to keep the graph on the screen.
z-axis
A second horizontal axis shows the standard normal distribution (the normal distribution with μ = 0 and
σ = 1) obtained by standardizing the current normal random variable.
{bml orig.bmp} A button to return to the starting situation.
$
Normale verdeling
Normaleverdeling
K Normal distribution
#
$ # K
Hypergeometric distribution
Drawing without replacement from a population that is composed of two subgroups generates a
hypergeometric distribution.
A hypergeometric distribution is defined by three parameters:
-the size of the population N
-the size of the subgroup D
-the size of the sample n.
P(X=k) is the probability that there are k successes (coming from the subgroup D) in the sample.
The probability distribution of the random variable X can be seen in the form of a bar chart or in the form
of a table.
In the table you see the cumulative probability P(X<=k) and the probability P(X>=k).
By clicking in a cell of the table you see the correspond part in red.
There are two settings.
Cumulative
The bar chart is drawn cumulative drawn. The color connection between table and graph is not active
now.
Normal approximation
The normal or cumulative normal distribution, which fits best, is drawn.
$
Hypergeometric distribution
Hypergeometrischeverdeling
K Hypergeometric distribution
#
$ # K
Poisson distribution
A Poisson distribution is defined by one parameter. This parameter has several names: lambda, the
mean, the rate, the intensity, and the density. A physical interpretation is the average number of events
during a time period of a specified duration.
With the Poisson distribution you can find the probability that a particular number of events will happen
during a time period.
The probability distribution of the random variable X you see in the form of a bar chart and in the form of
a table.
In the table you see the cumulative probability P(X>=k). By clicking in a cell of the table the
accompanying part of the graph is colored in red.
There are two settings.
Cumulative
The bar chart is drawn cumulative. . The color connection between table and graph is not active now.
Normal approximation
The normal or cumulative normal distribution, which fits best, is drawn.
$
Poisson verdeling
Poissonverdeling
K Poisson distribution
#
$ # K
Plots of distributions
With this option it is possible to compare different distributions, by comparing the graph or the table.
The distributions which can be used, are
-Binomial distribution
-Normal distribution
-Hyper geometric distribution
-Geometric distribution
-Exponential distribution
-Student’s t-distribution
-Chi-squared distribution
-F-distribution
There are a number of options and buttons:
Choose distribution
Choose a distribution and enter the parameters.
Filled rectangles
Option to fill the bars of a distribution in full color.
Add
The chosen distribution is added to the list and the graph is drawn.
Color
Select a color for the chosen distribution.
Axes
Set the bounds and intervals between the ticks for the axes.
Delete
To delete a selected distribution from the list.
Table
In the table you enter the x-value and the required precision for the probabilities.
bml openen.bmp} Open
To open a file containing distributions.
The file has extension .dst
{bml opslaan.bmp} Save
Save a file containing distributions
For a demonstration to a class you can prepare a file of distributions, and this avoids the trouble of typing
it in front of the class.
The file gets the extension .dst
$
Plots of distributions
Grafiekenvanverdelingen
K Plots of distributions
#
$ # K
Central Limit theorem
The central limit is presented with the aid of a die.
The probability of the sum or average of the dots a number of throws of the dices is calculated. Nearly
always the distribution converges to the normal distribution.
There is a number of settings and buttons
The central limit is demonstrated with the aid of a die.
The die is thrown a specified number of times and the probability distribution of the sum, or average, of
the scores on the die is calculated. The distribution always converges to the normal distribution as the
number of throws increases.
There are several options and buttons:
Number of throws
Probability 1/6
A fair die
Skewed left
A biased die. VU-Stat makes a suggestion, but you can enter the probabilities yourself if you wish.
Skewed right
A biased die. VU-Stat makes a suggestion, but you can enter the probabilities yourself if you wish.
Sum
The sum of the dots shown on the dice.
Average
The average of the dots shown on the dice.
Normal distribution
The best fitting normal distribution is shown.
$
Central Limit theorem
CentraleLimietstelling
K Central Limit theorem
#