Random variables and probabilities

Simple random variables

We consider, in some detail, random variables which have only a finite set of possible values. These are called simple random variables. Thus the term “simple” is used in a special, technical sense. The importance of simple random variables rests on two facts.For one thing, in practice we can distinguish only a finite set of possible values for any random variable. In addition, any random variable may be approximated as closely as pleased by a simplerandom variable. When the structure and properties of simple random variables have been examined, we turn to more general cases. Many properties of simple random variables extend tothe general case via the approximation procedure.

Representation with the aid of indicator functions

In order to deal with simple random variables clearly and precisely, we must find suitable ways to express them analytically. We do this with the aid of indicator functions . Three basic forms of representation are encountered. These are not mutually exclusive representatons.

1. Standard or canonical form , which displays the possible values and the corresponding events. If X takes on distinct values
   $$\{t_1,t_2,\cdots ,t_n\}\text{with}\text{respective}\text{probabilities}\{p_1,p_2,\cdots ,p_n\}$$
   and if $A_i=\{X=t_i\}$ , for $1\le i\le n$ , then $\{A_1,A_2,\cdots ,A_n\}$ is a partition (i.e., on any trial, exactly one of these events occurs). We call this the partition determined by (or, generated by) X . We may write
   $$X=t_1{I}_{A_1}+t_2{I}_{A_2}+\cdots +t_n{I}_{A_n}=\sum _{i=1}^n{t}_i{I}_{A_i}$$
   If $X\left(\omega \right)=t_i$ , then $\omega \in A_i$ , so that $I_{A_i}\left(\omega \right)=1$ and all the other indicator functions have value zero. The summation expression thus picks out the correct value t _i . This is true for any t _i , so the expression represents $X\left(\omega \right)$ for all ω . The distinct set $\{t_1,t_2,\cdots ,t_n\}$ of the values and the corresponding probabilities $\{p_1,p_2,\cdots ,p_n\}$ constitute the distribution for X . Probability calculations for X are made in terms of its distribution. One of the advantages of the canonical form is that it displays the range (set of values), and if the probabilities $p_i=P\left(A_i\right)$ are known, the distribution is determined. Note that in canonical form, if one of the t _i has value zero, we include that term. For some probability distributions it may be that $P\left(A_i\right)=0$ for one or more of the t _i . In that case, we call these values null values , for they can only occur with probability zero, and hence are practically impossible. Inthe general formulation, we include possible null values, since they do not affect any probabilitiy calculations.

   Successes in bernoulli trials

   As the analysis of Bernoulli trials and the binomial distribution shows (see Section 4.8), canonical form must be

   $$S_n=\sum _{k=0}^{n}k{I}_{A_k}\text{with}P\left(A_k\right)=C(n,k)p^k{(1-p)}^{n-k},0\le k\le n$$

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   For many purposes, both theoretical and practical, canonical form is desirable. For one thing, it displays directly the range (i.e., set of values) of the random variable. The distribution consists of the set of values $\{t_k:1\le k\le n\}$ paired with the corresponding set of probabilities $\{p_k:1\le k\le n\}$ , where $p_k=P\left(A_k\right)=P(X=t_k)$ .
2. Simple random variable X may be represented by a primitive form
   $$X=c_1{I}_{C_1}+c_2{I}_{C_2}+\cdots ,c_m{I}_{C_m},\text{where}\{C_j:1\le j\le m\}\text{is}\text{a}\text{partition}$$
   Remarks
   • If $\{C_j:1\le j\le m\}$ is a disjoint class, but ${\displaystyle \bigcup _{j=1}^m{C}_j\ne \Omega }$ , we may append the event ${\displaystyle C_{m+1}={\left[\bigcup _{j=1}^m,C_j\right]}^c}$ and assign value zero to it.
   • We say a primitive form, since the representation is not unique. Any of the C _i may be partitioned, with the same value c _i associated with each subset formed.
   • Canonical form is a special primitive form. Canonical form is unique, and in many ways normative.

   Simple random variables in primitive form

   • A wheel is spun yielding, on a equally likely basis, the integers 1 through 10. Let C _i be the event the wheel stops at i , $1\le i\le 10$ . Each $P\left(C_i\right)=0.1$ . If the numbers 1, 4, or 7 turn up, the player loses ten dollars; if the numbers 2, 5,or 8 turn up, the player gains nothing; if the numbers 3, 6, or 9 turn up, the player gains ten dollars; if the number 10 turns up, the player loses one dollar. The randomvariable expressing the results may be expressed in primitive form as
     $$X=-10I_{C_1}+0I_{C_2}+10I_{C_3}-10I_{C_4}+0I_{C_5}+10I_{C_6}-10I_{C_7}+0I_{C_8}+10I_{C_9}-I_{C_{10}}$$
   • A store has eight items for sale. The prices are $3.50, $5.00, $3.50, $7.50, $5.00, $5.00, $3.50, and $7.50, respectively. A customer comes in. She purchasesone of the items with probabilities 0.10, 0.15, 0.15, 0.20, 0.10 0.05, 0.10 0.15. The random variable expressing the amount of her purchase may be written
     $$X=3.5I_{C_1}+5.0I_{C_2}+3.5I_{C_3}+7.5I_{C_4}+5.0I_{C_5}+5.0I_{C_6}+3.5I_{C_7}+7.5I_{C_8}$$

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3. We commonly have X represented in affine form , in which the random variable is represented as an affine combination of indicator functions (i.e., a linearcombination of the indicator functions plus a constant, which may be zero).
   $$X=c_0+c_1{I}_{E_1}+c_2{I}_{E_2}+\cdots +c_m{I}_{E_m}=c_0+\sum _{j=1}^m{c}_j{I}_{E_j}$$
   In this form, the class $\{E_1,E_2,\cdots ,E_m\}$ is not necessarily mutually exclusive, and the coefficients do not display directly the set of possible values.In fact, the E _i often form an independent class. Remark . Any primitive form is a special affine form in which $c_0=0$ and the E _i form a partition.

   Consider, again, the random variable S _n which counts the number of successes in a sequence of n Bernoulli trials. If E _i is the event of a success on the i th trial, then one natural way to express the count is

   $$S_n=\sum _{i=1}^n{I}_{E_i},\text{with}P\left(E_i\right)=p1\le i\le n$$

   This is affine form, with $c_0=0$ and $c_i=1$ for $1\le i\le n$ . In this case, the E _i cannot form a mutually exclusive class, since they form an independent class.

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   Events generated by a simple random variable: canonical form
   We may characterize the class of all inverse images formed by a simple random X in terms of the partition $\{A_i:1\le i\le n\}$ it determines. Consider any set M of real numbers. If t _i in the range of X is in M , then every point $\omega \in A_i$ maps into t _i , hence into M . If the set J is the set of indices i such that $t_i\in M$ , then
   Only those points ω in ${\displaystyle A_M=\underset{i\in J}{\overset{}{\bigvee }}{A}_i}$ map into M .
   Hence, the class of events (i.e., inverse images) determined by X consists of the impossible event ∅ , the sure event Ω , and the union of any subclass of the A _i in the partition determined by X .

   Events determined by a simple random variable

   Suppose simple random variable X is represented in canonical form by

   $$X=-2I_A-I_B+0I_C+3I_D$$

   Then the class $\{A,B,C,D\}$ is the partition determined by X and the range of X is $\{-2,-1,0,3\}$ .

   1. If M is the interval $[-2,1]$ , then the values -2, -1, and 0 are in M and $X^{-1}\left(M\right)=A\bigvee B\bigvee C$ .
   2. If M is the set $(-2,-1]\cup [1,5]$ , then the values -1, 3 are in M and $X^{-1}\left(M\right)=B\bigvee D$ .
   3. The event $\{X\le 1\}=\{X\in (-\infty ,1]\}=X^{-1}\left(M\right)$ , where $M=(-\infty ,1]$ . Since values -2, -1, 0 are in M , the event $\{X\le 1\}=A\bigvee B\bigvee C$ .

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