GSOE9830 Chap.11 Expected Value, Variance and Risk Decisions
Expected Value, Variance and Risk Decisions
Expected value is the probability-weighted average of possible outcomes and is commonly used to roll back chance branches in a decision tree. Variance and standard deviation describe dispersion around that centre. The expected value is not the outcome guaranteed from one project, and equal means can hide very different downside exposure.
This chapter calculates all three measures, preserves negative outcomes and connects the numbers to risk capacity and decision limits. A deviation table centres every outcome on the same expected value, squares the deviation and weights it by probability before summing. Variance has squared units, while standard deviation returns to the original unit for interpretation.
Decision trees are rolled back from right to left, averaging only at chance nodes and choosing at decision nodes. A complete recommendation also considers downside thresholds and distribution shape rather than treating a higher mean as sufficient.
What this chapter covers
- 01
Weighted mean
- 02
Squared deviation
- 03
Variance
- 04
Standard deviation
- 05
Decision-tree rollback
- 06
Risk capacity
Worked example · free
Calculate expected savings and dispersion
- +1Expected value is $40,000.
- +1Weighted squared deviations sum to 200,000,000 dollars squared.
- +1Standard deviation is $14,142.14.
- +1Interpret the mean together with the spread.
Key terms
- Expected value
- The probability-weighted average of all stated outcomes.
- Variance
- The probability-weighted squared dispersion around expected value.
- Standard deviation
- The square root of variance expressed in the outcome's original unit.
- Decision rollback
- Evaluating a decision tree from later branches back toward the first choice.
Expected Value, Variance and Risk Decisions FAQ
Is expected value the most likely outcome?
Not necessarily. It is a probability-weighted centre and may not equal any listed outcome. It is most useful across repeated decisions or as a consistent comparison measure.
Why calculate standard deviation?
Variance is in squared units. Standard deviation returns dispersion to the same unit as the outcomes, making the scale easier to compare with expected value and decision limits.
How do I roll back a decision tree?
Start at the far-right chance nodes, calculate their expected values, and move left. At each decision node select the branch that best satisfies the stated economic criterion and constraints.
Exam move
Validate the distribution before calculating a centre. List every outcome with its sign and probability, confirm probabilities are non-negative and sum to one, and keep outcome units visible. Multiply each outcome by its probability in a separate column. Add the products to obtain expected value.
A negative outcome must remain negative through the multiplication, and an expected dollar result must be reported in dollars rather than as a probability. Interpret expected value as a probability-weighted long-run centre, not the amount guaranteed from one project. In a decision tree, work from right to left. Roll back each chance node by multiplying branch outcomes by conditional probabilities and summing.
At a decision node, apply the stated choice criterion rather than averaging alternatives the decision maker can choose between. Carry the selected branch value leftward and keep chance and decision nodes visually distinct. Calculate variance through a transparent deviation table. Subtract the same expected value from every outcome, square each deviation, multiply by probability and add. Record that variance has squared units.
Take the square root only after the weighted squared deviations have been summed; standard deviation then returns to the original outcome unit and is easier to interpret alongside the mean. Verify variance with the second-moment identity when the numbers permit. Compute the probability-weighted squared outcomes and subtract the square of expected value. Both routes should agree apart from rounding.
A negative variance signals arithmetic error, usually caused by an incorrect mean, premature rounding or a missing square. Recheck probability mass before trying to repair the final number. Compare alternatives carefully. Equal expected values do not imply equal risk when one distribution has greater spread or more severe downside.
A larger standard deviation is not automatically unacceptable: assess loss capacity, required reliability, risk preference and any hard threshold. Also inspect shape, because two distributions can share mean and standard deviation while placing loss probability differently. Practise a downside probability beside the summary measures.
Calculate the chance outcome falls below a limit and explain how that event changes the recommendation. This links a distribution to an engineering constraint more directly than quoting variance alone. If probabilities are estimates, note which branch probability deserves sensitivity testing.
Maintain an error log for probability total unchecked, loss sign dropped, expected value called guaranteed, deviations centred on different means, square root taken too early, squared units reported as dollars and decision branches averaged. Before submission, show the weighted products, mean, deviation table, variance, standard deviation and decision interpretation.
Finish by distinguishing the long-run centre from the single outcome that will actually occur and naming the downside condition most relevant to the choice.