MATH2070 Chap.7 Probability, Expectation and Covariance
Probability, Expectation and Covariance
Probability review supplies the moments used throughout financial mathematics. Expectation is a probability-weighted centre and is linear even for dependent variables. Variance measures squared spread, standard deviation restores the original return units, and covariance records paired movement. Correlation standardises covariance between minus one and one.
Joint state information matters because portfolio risk contains a covariance cross term that cannot be recovered from separate asset variances. A state table keeps probabilities and paired outcomes aligned, making the sign of covariance visible before calculation. The variance of a linear combination contains squared coefficients and doubled cross-products, so it cannot be obtained by simply averaging standard deviations.
Independence implies zero covariance when moments exist, but zero covariance alone need not imply independence. Consistent decimal or percentage-point units are essential because variance and covariance carry squared or combined scales while correlation is unit-free.
What this chapter covers
- 01
Discrete distributions
- 02
Expected values
- 03
Variance
- 04
Standard deviation
- 05
Covariance
- 06
Correlation
- 07
Linear combinations
Worked example · free
Compute moments for paired returns
- 1Check probabilities sum to one.
- 1Compute both expected returns.
- 1Compute one variance.
- 1Compute and interpret covariance.
Key terms
- Expected value
- The probability-weighted average of a random variable's possible outcomes.
- Variance
- The expected squared distance of a random variable from its mean.
- Covariance
- The expected product of paired deviations from two means.
- Correlation
- Covariance divided by both standard deviations, producing a unit-free dependence measure.
- Joint state
- One simultaneous outcome for all random variables in a model, paired with its probability. Joint states preserve the outcome matching required for covariance and cannot be reconstructed from separate marginal lists alone.
- Second moment
- The expectation of a squared random variable. Combined with the squared mean through variance equals second moment minus mean squared, it provides an alternative calculation that should agree with squared deviations. Keeping the mean unrounded helps prevent a small negative result caused purely by premature arithmetic rounding.
Probability, Expectation and Covariance FAQ
Does expected return predict the realised return?
No. It is the probability-weighted centre of the distribution and may not be an attainable outcome. Realised return depends on which state occurs, while expectation summarises repeated or modelled uncertainty.
Why is covariance needed for portfolio risk?
Separate variances describe each asset's spread but not whether their deviations occur together. Covariance supplies that joint movement and can raise or lower the variance of a weighted combination.
Does zero correlation mean independence?
Not generally. Independence implies zero covariance when moments exist, but nonlinear dependence can produce zero covariance or correlation. The reverse conclusion needs additional distributional assumptions.
Exam move
Build a state table before using a formula. Give each row a state, probability and outcome, and add paired outcomes when covariance is required. Confirm that probabilities are non-negative and sum to one. Compute the mean in a dedicated column, then write deviations beside their states instead of performing all products on a calculator. This layout exposes a misplaced probability and keeps joint outcomes aligned.
Practise variance by both definitions. Use the weighted squared-deviation form to see why the result cannot be negative, and use the second-moment shortcut to check the same answer. Keep the mean unrounded until subtraction is complete. After obtaining variance, label its squared units and take a square root only when standard deviation is requested.
Alternate decimal-return and percentage-point problems so that conversion is a conscious final step rather than a hidden mixture of scales. For covariance, place the two deviations on the same state row, multiply them and inspect their signs before summing. Predict positive or negative covariance from the state pattern, then compare the calculation with that prediction.
Convert covariance to correlation only after calculating both standard deviations, and check that the result lies between minus one and one. Include examples where covariance is zero but the variables are not independent so that the logical distinction remains clear. Use short retrieval sets combining expectation linearity, variance of a linear combination and covariance.
State explicitly when independence is unnecessary, when it implies a zero covariance term, and when the supplied covariance must be retained. Finish finance applications by converting results back to expected return and risk language. Record errors under probability setup, units, mean, deviation pairing or formula expansion.
A weekly mixed problem should be solved first symbolically, then numerically, with a final reasonableness check on the mean range, variance sign and correlation bound.
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