Random variables and probabilities (Page 5/8)

Applied probability Page 5 / 8

Determination of the distribution

Determining the partition generated by a simple random variable amounts to determining the canonical form. The distribution is then completed by determining the probabilities ofeach event $A_{k} = {X = t_{k}}$ .

From a primitive form

Before writing down the general pattern, we consider an illustrative example.

The distribution from a primitive form

Suppose one item is selected at random from a group of ten items. The values (in dollars) and respective probabilities are

c _j	2.00	1.50	2.00	2.50	1.50	1.50	1.00	2.50	2.00	1.50
$P (C_{j})$	0.08	0.11	0.07	0.15	0.10	0.09	0.14	0.08	0.08	0.10

By inspection, we find four distinct values: $t_{1} = 1.00$ , $t_{2} = 1.50$ , $t_{3} = 2.00$ , and $t_{4} = 2.50$ . The value 1.00 is taken on for $ω \in C_{7}$ , so that $A_{1} = C_{7}$ and $P (A_{1}) = P (C_{7}) = 0.14$ . Value 1.50 is taken on for $ω \in C_{2}, C_{5}, C_{6}, C_{10}$ so that

A_{2} = C_{2} ⋁ C_{5} ⋁ C_{6} ⋁ C_{10} and P (A_{2}) = P (C_{2}) + P (C_{5}) + P (C_{6}) + P (C_{10}) = 0.40

Similarly

P (A_{3}) = P (C_{1}) + P (C_{3}) + P (C_{9}) = 0.23 and P (A_{4}) = P (C_{4}) + P (C_{8}) = 0.23

The distribution for X is thus

k	1.00	1.50	2.00	2.50
$P (X = k)$	0.14	0.40	0.23	0.23

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The general procedure may be formulated as follows:

If $X = \sum_{j = 1}^{m} c_{j} I_{C_{j}}$ , we identify the set of distinct values in the set ${c_{j} : 1 \leq j \leq m}$ . Suppose these are $t_{1} < t_{2} < \dots < t_{n}$ . For any possible value t _i in the range, identify the index set J _i of those j such that $c_{j} = t_{i}$ . Then the terms

\sum_{J_{i}} c_{j} I_{C_{j}} = t_{i} \sum_{J_{i}} I_{C_{j}} = t_{i} I_{A_{i}}, where A_{i} = ⋁_{j \in J_{i}}^{} C_{j},

and

P (A_{i}) = P (X = t_{i}) = \sum_{j \in J_{i}} P (C_{j})

Examination of this procedure shows that there are two phases:

Select and sort the distinct values $t_{1}, t_{2}, \dots, t_{n}$
Add all probabilities associated with each value t _i to determine $P (X = t_{i})$

We use the m-function csort which performs these two operations (see Example 4 from "Minterms and MATLAB Calculations").

Use of csort on [link]

>>C = [2.00 1.50 2.00 2.50 1.50 1.50 1.00 2.50 2.00 1.50];  % Matrix of c_j>>pc = [0.08 0.11 0.07 0.15 0.10 0.09 0.14 0.08 0.08 0.10]; % Matrix of P(C_j)>>[X,PX] = csort(C,pc);     % The sorting and consolidating operation>>disp([X;PX]')             % Display of results1.0000    0.1400
    1.5000    0.40002.0000    0.2300
    2.5000    0.2300

For a problem this small, use of a tool such as csort is not really needed. But in many problems with large sets of data the m-function csort is very useful.

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From affine form

Suppose X is in affine form,

X = c_{0} + c_{1} I_{E_{1}} + c_{2} I_{E_{2}} + \dots + c_{m} I_{E_{m}} = c_{0} + \sum_{j = 1}^{m} c_{j} I_{E_{j}}

We determine a particular primitive form by determining the value of X on each minterm generated by the class ${E_{j} : 1 \leq j \leq m}$ . We do this in a systematic way by utilizing minterm vectors and properties of indicator functions.

X is constant on each minterm generated by the class ${E_{1}, E_{2}, \dots, E_{m}}$ since, as noted in the treatment of the minterm expansion, each indicator function $I_{E_{i}}$ is constant on each minterm. We determine the value s _i of X on each minterm M _i . This describes X in a special primitive form
$X = \sum_{k = 0}^{2^{m} - 1} s_{i} I_{M_{i}}, with P (M_{i}) = p_{i}, 0 \leq i \leq 2^{m} - 1$
We apply the csort operation to the matrices of values and minterm probabilities to determine the distribution for X .

We illustrate with a simple example. Extension to the general case should be quite evident. First, we do the problem “by hand” in tabular form. Then we use them-procedures to carry out the desired operations.

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Source: OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6

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