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What is a probability distribution?

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Jones Kalunga This work is produced by OpenStax-CNX and licensed under the † Creative Commons Attribution License 2.0

Denition 1: Pobability Distribution

A probability distribution Pr{} on a sample space S is a mapping from events of S to real numbers.[1] In general, Probability Distributions are classied into two categories: : If the distribution function is for a discrete random variable2 X as in the example below, then the function is called discrete. [3] • Continuous Probability Distribution3 : If the distribution function is for a continuous random variable4 X, then the function is called continuous. [3]

• Discrete Probability Distribution

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Probability distributions are a fundamental concept in statistics. They are used both on a theoretical level and a practical level. Some practical uses of probability distributions are: • To calculate condence intervals for parameters and to calculate critical regions for hypothesis tests. • For univariate data, it is often useful to determine a reasonable distributional model for the data. • Statistical intervals and hypothesis tests are often based on specic distributional assumptions. Before

computing an interval or test based on a distributional assumption, we need to verify that the assumption is justied for the given data set. In this case, the distribution does not need to be the best-tting distribution for the data, but an adequate enough model so that the statistical technique yields valid conclusions. • Simulation studies with random numbers generated from using a specic probability distribution are often needed.[2]

Example 1

Suppose that two of fair dice are tossed. This time, let the random variable X denote the sum of the points. What is the sample space and what is the probability distribution for this experiment? [3] In the Sample Space below, the rst number of the ordered pair is the number showing on the rst die, and the second number is the number showing on the second die. Notice that there are thirty-six possible results so the sample space has thirty-six elements. ∗ Version

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1 http://www.itl.nist.gov/div898/handbook/eda/section3/eda361.htm 2 http://cne.gmu.edu/modules/dau/prob/randomvars/drv_frm.html 3 http://www.itl.nist.gov/div898/handbook/eda/section3/eda361.htm 4 http://cne.gmu.edu/modules/dau/prob/randomvars/crv_frm.html

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Table: Sample space (Redrawn from cne.gmu.edu/modules/dau/prob/distributions/dis_1_frm.html example 2 probabilty distributions) (1,6)

(2,6)

(3,6)

(4,6)

(5,6)

(6,6)

(1,5)

(2,5)

(3,5)

(4,5)

(5,5)

(6,5)

(1,4)

(2,4)

(3,4)

(4,4)

(5,4)

(6,4)

(1,3)

(2,3)

(3,3)

(4,3)

(5,3)

(6,3)

(1,2)

(2,2)

(3,2

(4,2)

(5,2)

(6,2)

(1,1)

(2,1)

(3,1)

(4,1)

(5,1)

(6,1)

Table 1

In the Probability Distribution Table below, X is the sum of the two numbers showing on the dice. If X = 2, the number showing on the rst die must be one and the second die also is one. The distribution table shows there is only one chance out of thirty-six that both dice show one. When X = 3, the rst die shows 1 and the second die shows 2 or vice versa. Thus there are two chances in thirty-six of this happening.

Table: Probability distribution table (Redrawn from cne.gmu.edu/modules/dau/prob/distributions/dis_1_frm.html example 2 probabilty distributions) x 2 3 4 5 6 7 8 9 10 11 12 f(x) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 Table 2

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Probability Distribution

Figure 1:

A matlab generated graph of the probability distribution from the table above.

Exercise 1

(Solution on p. 5.)

Suppose that a coin is tossed twice so that the sample space S = {HH, HT, TH, TT}. Let X be the number of heads which can come up. What is the associated probability distribution for the variable X [3] ?

1 References: 1. Introduction to Algorithms, http://highered.mcgraw-hill.com/sites/0070131511/student_view0/glossary_ps.html5 (last accessed 21 Febraury 2006) 2. Engineering Statistics Handbook, www.itl.nist.gov/div898/handbook/eda/section3/eda36.htm6 (last accessed 21 Febraury 2006) 3. Amar B. Rao, Priscilla McAndrews. /et al. Defence Acquisition University Statistics Refresher Module. , http://cne.gmu.edu/modules/dau/prob/distributions/dis_1_frm.html7 (last accessed 21 Febraury 2006) 5 http://highered.mcgraw-hill.com/sites/0070131511/student_view0/glossary_p-s.html 6 http://cnx.org/content/m13464/latest/www.itl.nist.gov/div898/handbook/eda/section3/eda36.htm 7 http://cne.gmu.edu/modules/dau/prob/distributions/dis_1_frm.html

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Co-Author: Mookho Tsilo

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Solutions to Exercises in this Module Solution to Exercise (p. 3) Thus, in the case of HH (2 heads), X= 2 while for TH (1 head), X = 1. Table: Probability distribution of the experiment (redrawn from cne.gmu.edu/modules/dau/prob/distributions/dis_1_frm.html) Sample Point HH HT TH TT X 2 1 1 0 Table 3

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