MFIN6201 Chap.1 Probability, Dependence and Conditioning
Probability, Dependence and Conditioning
This chapter turns uncertain financial outcomes into distributions that can be analysed. It covers expected value, variance, standard deviation, joint and conditional distributions, covariance, correlation and independence. The key discipline is logical direction: independence implies zero covariance, but zero covariance alone does not establish independence.
Worked calculations keep probability weights and return units visible. Begin with the sample space and attach a probability to each outcome before multiplying. The expected value balances the distribution but need not be an outcome that can occur. Variance squares deviations from that centre, while standard deviation returns the answer to the variable's original units.
With two variables, the joint distribution retains combinations, a marginal distribution sums over the other variable, and a conditional distribution updates after information arrives. Covariance supplies the cross term in portfolio variance, so its sign explains diversification. Correlation standardises covariance but describes linear association only.
Conditional means provide a bridge to regression: later, conditional mean zero will protect an OLS slope. The law of iterated expectations checks the conditioning work by averaging group means back to the population mean. Every practice answer should name whether it uses a marginal, conditional or joint probability and should stop short of reversing a one-way implication.
What this chapter covers
- 01
Expected values from discrete states
- 02
Variance and standard deviation
- 03
Joint and conditional distributions
- 04
Covariance and correlation
- 05
Independence and mean independence
- 06
Law of iterated expectations
Worked example · free
Expected return across three states
- +1Verify probabilities sum to one: 0.25+0.50+0.25=1.
- +1Compute 0.25(-0.02)+0.50(0.04)+0.25(0.09)=0.0375.
- +1Interpret 0.0375 as an expected return of 3.75%.
Key terms
- Expected value
- A probability-weighted average of possible outcomes.
- Covariance
- A measure of how two variables' deviations from their means move together.
- Mean independence
- A condition under which a conditional mean does not vary with the conditioning variable.
Probability, Dependence and Conditioning FAQ
Why does zero correlation not prove independence?
Correlation measures linear dependence only. Variables can have a strong nonlinear relationship whose positive and negative linear components cancel, leaving correlation at zero. Independence is a broader distributional claim and therefore requires more than a covariance or correlation calculation.
When should I use conditional expectation?
Use it when new information divides outcomes into meaningful states or groups. Compute the mean within each conditioning state first, then use the law of iterated expectations to average those conditional means and recover the unconditional expectation.
How do I check a variance calculation?
Confirm every probability is attached to the correct squared deviation, probabilities sum to one, and the result is nonnegative. Standard deviation should be the square root of variance and return to the original units of the random variable.
What does covariance add to portfolio analysis?
Individual variances describe standalone spread, while covariance records whether two assets move around their means together. Its weighted cross term can lower portfolio variance when negative, but an estimated historical sign does not guarantee stable dependence during stress.
How does iterated expectation check my work?
Compute the mean within each conditioning group, weight those group means by group probabilities, and add them. The result should equal the unconditional mean. A mismatch usually reveals an incorrect conditional probability or group weight.
Why do units matter for variance?
Variance is expressed in squared units, so it cannot be compared directly with the mean in the original scale. Taking the square root produces standard deviation, which returns to return, dollar or score units and supports a clearer interpretation. Always name that original scale when reporting the result.
Can conditional means differ when the unconditional mean is unchanged?
Yes. Different groups can have offsetting conditional means whose probability-weighted average equals the same unconditional centre. Conditioning reveals that heterogeneity; the overall average alone can hide it and should not be treated as every group's expected outcome. Always reconcile the group weights before comparing those two summaries.
Exam move
Create a compact implication map: independence implies mean independence, which implies zero covariance, but the reverse arrows generally fail. Recompute moments from both the squared-deviation and second-moment identities. Use conditional expectation problems to practise changing probability weights after information arrives, and finish every portfolio calculation by explaining the covariance sign.
Build a three-column worksheet for each problem: state, probability and weighted contribution. Add a fourth column for squared deviations only after the mean is settled. For two-variable questions, draw a small joint table and recover both marginals before conditioning; this prevents denominators from drifting. Test the law of iterated expectations numerically by recombining the conditional means.
Practise one nonlinear counterexample so zero correlation never becomes a synonym for independence. When revising portfolio variance, label each variance term and the covariance cross term with its coefficient weight. Use percentage points for interpretation but decimals for multiplication. Finish by writing one sentence on what new information changes the distribution and another on what remains unproved.
This chapter rewards clean logic as much as arithmetic, so check implication direction before checking the calculator.
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