ECMT1020 Chap.3 Cumulative Probability and Expected Value
Cumulative Probability and Expected Value
The cumulative distribution function collects probability from the lower tail up to a threshold and works for both discrete and continuous variables. This chapter uses differences of cumulative values to calculate interval probabilities, then develops expectations as probability-weighted averages.
It also applies the law of the unconscious statistician to transformations, keeping the original distribution while changing the function being averaged. A cumulative function must start near zero, never decrease and approach one as the threshold moves beyond the support. In a discrete case it rises in jumps whose sizes equal point masses; in a continuous case it accumulates smoothly when the density is well behaved.
Expectations answer a different question. They balance every possible value using probability weights and can be calculated for the variable itself or for a transformation such as a square, cost or indicator. Keeping cumulative probability and weighted averaging conceptually separate makes their shared use of the distribution much clearer.
What this chapter covers
- 01
Cumulative distribution functions
- 02
Discrete jumps
- 03
Continuous cumulative probability
- 04
Interval probabilities
- 05
Expected values
- 06
Expectations of transformations
Worked example · free
Use a cumulative distribution
- +1For a continuous variable, subtract cumulative values at the endpoints.
- +1The probability is F(5)-F(2)=0.80-0.25=0.55.
- +1A discrete calculation must inspect whether mass at the lower endpoint is included.
- +1The interval symbols and jumps of the cumulative function determine that endpoint treatment.
Key terms
- Cumulative distribution
- The probability that a variable does not exceed a threshold.
- Lower tail
- The set of values at or below a chosen point.
- Expected value
- The distribution-weighted average of a random variable.
- Transformation
- A function applied to each possible value of a variable.
- Indicator variable
- A variable equal to one when an event occurs and zero otherwise.
- Right continuity
- The cumulative-distribution property that its value at a threshold includes any mass located exactly there and agrees with values approached from the right. It determines how jumps are drawn for a discrete random variable.
- Weighted centre
- A summary formed by multiplying every supported value by its probability and adding the products. It describes the long-run balance point of the distribution and need not itself be a possible realised value.
- Cumulative jump
- The vertical increase of a discrete cumulative distribution at a supported value. Its size equals the probability mass assigned to that exact value.
Cumulative Probability and Expected Value FAQ
Why must a cumulative distribution increase?
Raising the threshold can only add possible outcomes to the event X≤x. It cannot remove probability already accumulated, so the cumulative function never decreases.
How do I find an interval probability?
For a continuous variable, subtract the cumulative value at the lower endpoint from the value at the upper endpoint. For discrete variables, inspect endpoint inclusion and jumps.
Must an expected value be possible?
No. An expectation is a weighted centre and need not belong to the support. A count variable can have a non-integer expectation even though every realised count is an integer.
Exam move
Sketch cumulative functions as accumulated probability, not as a second density. Practise moving between a mass or density description and its cumulative form. For expectations, write each supported value and its probability before multiplying. When transforming a variable, keep the original probabilities attached to their corresponding transformed values. Revise cumulative calculations with a boundary table.
Give each row an interval, the cumulative value at its upper edge and any jump at that edge. Then answer probability questions by marking which mass belongs inside the interval before subtracting. This makes strict and weak inequalities visible instead of leaving endpoint treatment to memory. Draw discrete cumulative functions as right-continuous staircases and label each jump with its probability mass.
For continuous examples, derive the cumulative function piecewise, including zero below the support and one above it. Differentiate the middle piece as a check that it returns the density. For expectation drills, prepare a value, probability, product table and add an optional transformation column. Compute the raw expectation first, then replace each value with the transformed value while leaving its probability unchanged.
This prevents the common mistake of inventing a new distribution for g(X). Use indicator variables to connect the ideas: the expectation of an event indicator equals the event probability, so a probability can be written as a special expected value. Add interpretation after arithmetic. An expected profit is a long-run probability-weighted centre, not a guaranteed outcome; it can even fall between supported values.
Compare two prospects only after checking that the same variable and time horizon are being averaged. Your error log should distinguish an omitted support value, an incorrect cumulative subtraction, a lost endpoint mass, a transformation error and a multiplication slip. Repair each category with fresh values.
On a final pass, confirm that the cumulative answer lies from zero to one, the function is nondecreasing, and the expectation uses every supported value exactly once.
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