Mathematical expectation: simple random variables (Page 3/4)

Applied probability Page 3 / 4

Matlab calculation for example 3

X = [2 2.5 3 3.5];  % Matrix of values (ordered)PX = 0.1*[2 2 3 3]; % Matrix of probabilitiesEX = dot(X,PX)      % The usual MATLAB operation
EX =  2.8500Ex = sum(X.*PX)     % An alternate calculation
Ex =  2.8500ex = X*PX'          % Another alternate
ex =  2.8500

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Expectation and primitive form

The definition and treatment above assumes X is in canonical form , in which case

X = \sum_{i = 1}^{n} t_{i} I_{A_{i}}, where A_{i} = {X = t_{i}}, implies E [X] = \sum_{i = 1}^{n} t_{i} P (A_{i})

We wish to ease this restriction to canonical form.

Suppose simple random variable X is in a primitive form

X = \sum_{j = 1}^{m} c_{j} I_{C_{j}}, where {C_{j} : 1 \leq j \leq m} is a partition

We show that

E [X] = \sum_{j = 1}^{m} c_{j} P (C_{j})

Before a formal verification, we begin with an example which exhibits the essential pattern. Establishing the general case is simply a matter of appropriate use of notation.

Simple random variable X In primitive form

X = I_{C_{1}} + 2 I_{C_{2}} + I_{C_{3}} + 3 I_{C_{4}} + 2 I_{C_{5}} + 2 I_{C_{6}}, with {C_{1}, C_{2}, C_{3}, C_{4}, C_{5} . C_{6}} a partition

Inspection shows the distinct possible values of X to be 1, 2, or 3. Also,

A_{1} = {X = 1} = C_{1} ⋁ C_{3}, A_{2} = {X = 2} = C_{2} ⋁ C_{5} ⋁ C_{6} and A_{3} = {X = 3} = C_{4}

so that

P (A_{1}) = P (C_{1}) + P (C_{3}), P (A_{2}) = P (C_{2}) + P (C_{5}) + P (C_{6}), and P (A_{3}) = P (C_{4})

Now

E [X] = P (A_{1}) + 2 P (A_{2}) + 3 P (A_{3}) = P (C_{1}) + P (C_{3}) + 2 [P (C_{2}) + P (C_{5}) + P (C_{6})] + 3 P (C_{4})

= P (C_{1}) + 2 P (C_{2}) + P (C_{3}) + 3 P (C_{4}) + 2 P (C_{5}) + 2 P (C_{6})

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To establish the general pattern, consider $X = \sum_{j = 1}^{m} c_{j} I_{C_{j}}$ . We identify the distinct set of values containedin the set ${c_{j} : 1 \leq j \leq m}$ . Suppose these are $t_{1} < t_{2} < \dots < t_{n}$ . For any 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}

By the additivity of probability

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

Since for each $j \in J_{i}$ we have $c_{j} = t_{i}$ , we have

E [X] = \sum_{i = 1}^{n} t_{i} P (A_{i}) = \sum_{i = 1}^{n} t_{i} \sum_{j \in J_{i}} P (C_{j}) = \sum_{i = 1}^{n} \sum_{j \in J_{i}} c_{j} P (C_{j}) = \sum_{j = 1}^{m} c_{j} P (C_{j})

— $□$

Thus, the defining expression for expectation thus holds for X in a primitive form .

An alternate approach to obtaining the expectation from a primitive form is to use the csort operation to determine the distribution of X from the coefficients and probabilities of the primitive form.

Alternate determinations of $E [X]$

Suppose X in a primitive form is

X = I_{C_{1}} + 2 I_{C_{2}} + I_{C_{3}} + 3 I_{C_{4}} + 2 I_{C_{5}} + 2 I_{C_{6}} + I_{C_{7}} + 3 I_{C_{8}} + 2 I_{C_{9}} + I_{C_{10}}

with respective probabilities

P (C_{i}) = 0.08, 0.11, 0.06, 0.13, 0.05, 0.08, 0.12, 0.07, 0.14, 0.16

c = [1 2 1 3 2 2 1 3 2 1];            % Matrix of coefficientspc = 0.01*[8 11 6 13 5 8 12 7 14 16]; % Matrix of probabilitiesEX = c*pc'
EX =   1.7800                         % Direct solution[X,PX] = csort(c,pc);                 % Determination of dbn for Xdisp([X;PX]')1.0000    0.4200
    2.0000    0.38003.0000    0.2000
Ex = X*PX'                            %  E[X]from distribution
Ex =   1.7800

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Linearity

The result on primitive forms may be used to establish the linearity of mathematical expectation for simple random variables. Because of its fundamental importance, we work throughthe verification in some detail.

Suppose $X = \sum_{i = 1}^{n} t_{i} I_{A_{i}}$ and $Y = \sum_{j = 1}^{m} u_{j} I_{B_{j}}$ (both in canonical form). Since

\sum_{i = 1}^{n} I_{A_{i}} = \sum_{j = 1}^{m} I_{B_{j}} = 1

we have

X + Y = \sum_{i = 1}^{n} t_{i} I_{A_{i}} (\sum_{j = 1}^{m}, I_{B_{j}}) + \sum_{j = 1}^{m} u_{j} I_{B_{j}} (\sum_{i = 1}^{n}, I_{A_{i}}) = \sum_{i = 1}^{n} \sum_{j = 1}^{m} (t_{i} + u_{j}) I_{A_{i}} I_{B_{j}}

Note that $I_{A_{i}} I_{B_{j}} = I_{A_{i} B_{j}}$ and $A_{i} B_{j} = {X = t_{i}, Y = u_{j}}$ . The class of these sets for all possible pairs $(i, j)$ forms a partition. Thus, the last summation expresses $Z = X + Y$ in a primitive form. Because of the result on primitive forms, above, we have

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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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Mathematical expectation: simple random variables (Page 3/4)

Matlab calculation for example 3

Simple random variable X In primitive form

Alternate determinations of E [ X ]

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Alternate determinations of $E [X]$