ECMT1020 Chap.5 Independence and Probability Calculations
Independence and Probability Calculations
Independence says that learning one variable does not change the distribution of another and permits joint probabilities to factor into marginal products. This chapter contrasts that strong property with zero covariance, which concerns only one linear product. It uses event enumeration, joint cells and conditional probabilities to test dependence, and shows why symmetry can hide a nonlinear relationship from correlation.
Independence is a statement about the entire joint distribution, so one matching product cell cannot establish it, while one failed cell is enough to disprove it. Conditional probabilities provide the same diagnostic from another direction: if the distribution of Y changes after X is known, dependence is present. Event calculations still require careful enumeration.
Factorisation may simplify the probability of each joint pair, but it never identifies which pairs satisfy the event for you.
What this chapter covers
- 01
Joint and marginal probabilities
- 02
Independence factorisation
- 03
Conditional probability
- 04
Independent functions
- 05
Zero covariance
- 06
Nonlinear dependence
Worked example · free
Construct dependence with zero covariance
- +1Y is determined by X, so dependence is plausible.
- +1P(X=2,Y=1)=0.25 but P(X=2)P(Y=1)=0.125.
- +1The failed cell proves the variables are dependent.
- +1Symmetry gives E(X)=0 and E(Y)=0.5.
- +1The negative and positive XY contributions cancel, so E(XY)=0.
- +1Cov(X,Y)=0 although the variables are dependent.
Key terms
- Independence
- A property under which joint probabilities factor across the full support.
- Marginal probability
- A probability for one variable after aggregating over the other.
- Joint probability
- The probability that specified values or events occur together.
- Conditional probability
- Probability updated after another event or value is known.
- Uncorrelated variables
- Variables with covariance equal to zero.
- Product rule
- The independence requirement that a joint probability equals the product of the corresponding marginal probabilities throughout the supported joint distribution. One failed comparison proves dependence, whereas one successful comparison is not sufficient.
- Event enumeration
- The process of listing every mutually exclusive supported outcome pair that satisfies a joint event before evaluating probabilities. It prevents valid combinations from being lost when independence makes each cell easy to multiply.
- Nonlinear dependence
- A relationship in which one variable changes systematically with another without a simple linear pattern. Such dependence can remain after covariance contributions cancel to zero.
Independence and Probability Calculations FAQ
Does independence imply zero correlation?
When the relevant moments and nonzero standard deviations exist, independence implies zero covariance and therefore zero correlation. The reverse implication fails in general for nonlinear relationships.
How can I disprove independence?
Find one supported joint event whose probability differs from the product of its marginals, or show that a conditional distribution changes with the conditioning value somewhere on the complete joint support of both variables.
When may I multiply marginal probabilities?
Only when independence is given or established for the events or variables involved. Otherwise use the joint distribution or a conditional-probability identity that preserves the dependence information needed for the event.
Exam move
Keep two proof templates distinct. For independence, compare joint probabilities with products of marginals across the support. For covariance, compute E(XY)-E(X)E(Y). Practise symmetric nonlinear examples until you can explain why cancellation makes covariance zero without removing dependence. Always list mutually exclusive event pairs before adding their probabilities. Build joint tables from the margins outward.
First verify that every cell is nonnegative and that the cells sum to one. Next calculate row and column totals independently. Only then compare each joint cell with the corresponding product of marginals. This order protects the test from a wrong margin. For conditional checks, divide a joint cell by the conditioning marginal and compare the resulting conditional distribution across conditioning values.
Translate the result into words about whether learning one variable changes beliefs about the other. Separate event enumeration from probability evaluation. If asked for a sum, product or threshold event, list every supported pair satisfying the condition before using independence. Mark the pairs as mutually exclusive and add their probabilities. This prevents multiplication from hiding an omitted combination.
Rehearse the implication direction with two examples. Use genuinely independent variables to show covariance factorisation, then construct a symmetric nonlinear relation in which positive and negative centred products cancel. Explain why the second pair remains dependent even though its covariance is zero. A transformation such as a square can reveal the concealed relationship.
Keep an error log with three labels: unauthorised multiplication, incomplete event list and reversed implication. For each entry, write the minimum evidence needed to repair it. Under time pressure, a concise proof of dependence needs only one failed product cell with correct marginals, while a claim of independence needs the full supported structure or an assumption supplied by the question.
Close every calculation by identifying whether the result concerns a joint event, conditional belief, covariance or the stronger property of independence.
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