A finite random-variable distribution
A finite random variable assigns a numerical value to each elementary outcome. Group all outcomes with the same value and add their exact probabilities to form the distribution. The binary illustration below uses the four equally likely ordered success/failure outcomes and counts successes
\[\Pr(X=x)=\sum_{\omega:\,X(\omega)=x}\Pr(\omega)\]
- Define the random variable Apply the counting rule to each ordered outcome in this equal-mass illustration \(\displaystyle \begin{gathered}\begin{gathered}S=\text{success},\quad F=\text{failure}\\[4pt]\Omega=\{(S,S),(S,F),(F,S),(F,F)\}\\[4pt]\Pr(\omega)=\frac14\quad(\omega\in\Omega)\\[4pt]X(\omega)=\text{number of successes}\end{gathered}\\[6pt]X(S,F)=1\end{gathered}\)
- Keep ordered outcomes Outcomes in different stage orders remain different elementary outcomes \(\displaystyle (S,F)\ne(F,S)\)
- Group equal values Add the masses of every outcome that produces the same value \(\displaystyle \Pr(X=1)=\Pr(S,F)+\Pr(F,S)=\frac{2}{4}\)
- Check the total mass The probabilities of all possible X values must add to one \(\displaystyle \sum_x\Pr(X=x)=\frac{1}{4}+\frac{1}{2}+\frac{1}{4}=1\)
Worked example
A separate experiment uses the unequal elementary probabilities shown. Apply the stated counting rule and find the probability that the count is at least one
\[\begin{gathered}\begin{array}{c|c}\omega&\Pr(\omega)\\\hline(A,A)&1/6\\(A,B)&1/3\\(B,A)&1/4\\(B,B)&1/4\end{array}\\[6pt]X(\omega)=\text{number of A outcomes}\\[6pt]\Pr(X\ge1)=?\end{gathered}\]
- \(\displaystyle \begin{gathered}X(A,A)=2\\X(A,B)=X(B,A)=1\\X(B,B)=0\end{gathered}\)Apply the definition of X to every ordered outcome
- \(\displaystyle \begin{gathered}\Pr(X=0)=\frac14\\[4pt]\Pr(X=1)=\frac13+\frac14=\frac7{12}\\[4pt]\Pr(X=2)=\frac16\end{gathered}\)Group the outcomes that have the same value of X and add their probabilities
- \(\displaystyle \begin{aligned}\Pr(X\ge1)&=\Pr(X=1)+\Pr(X=2)\\[4pt]&=\frac7{12}+\frac16=\frac9{12}=\frac34\end{aligned}\)Add the probabilities for the values of X that are at least 1
Common mistakes
Equal-mass experiment: counting each value once
✕\(\displaystyle \Pr(X=1)=\frac{1}{4}\)✓\(\displaystyle \begin{gathered}\Pr(S,F)=\Pr(F,S)=\frac14\\[4pt]\Pr(X=1)=\frac14+\frac14=\frac24=\frac12\end{gathered}\)
Here the four ordered success/failure outcomes are equally likely and the variable counts successes. The two mixed orders each contribute their elementary probabilityTreating the value as the outcome
✕\(\displaystyle X(A,B)=B\)✓\(\displaystyle X(A,B)=1\)
In the unequal-mass example, the rule counts occurrences of A. The ordered pair is the input and its count is the numerical valueUnequal-mass experiment: assigning one-quarter to every value
✕\(\displaystyle \Pr(X=0)+\Pr(X=1)+\Pr(X=2)=\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{4}\)✓\(\displaystyle \Pr(X=0)+\Pr(X=1)+\Pr(X=2)=\frac{1}{4}+\frac{7}{12}+\frac{1}{6}=1\)
Use the elementary probabilities from the separate unequal-mass example, grouping outcomes by their count. The complete distribution must total oneDifficulty levels
introMap a small binary experiment to a count-valued variable and complete its frequencies
standardComplete a probability distribution for a sum, count, maximum or minimum
challengeUse unequal elementary masses or a non-monotone lookup mapping
examRecognise a valid finite distribution and calculate the probability of a set of X values
Questions about this topic
What is a finite random variable?+
It is a rule that assigns one numerical value to each outcome in a finite sample space
Why are outcomes with the same value grouped?+
The distribution records the probability of the value, so the probabilities of all outcomes producing that value must be added
Why must the probabilities in the distribution total one?+
The listed values of the random variable must cover every possible outcome, so their probabilities together account for the whole sample space
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 8: Probability and counting
In this chapter
Finite events, probability assignments and complementsRepresent a multi-stage finite sample spaceCompound events, the addition rule and mutual exclusivityComplete and use a two-way probability tableConditional probability from exact representationsDistribution of a finite random variableDesign and interpret a probability simulationCount restricted selections and calculate exact probabilitiesCount arrangements with product, sum and restriction rulesTest exact independenceExact probability of repeated independent successesMulti-stage probability, multiplication and total probabilityProbability from a uniform area modelProbabilities of sequences of independent events