A theoretical weighted average
For an integrable discrete variable, expectation sums each possible value multiplied by its probability. This theoretical mean differs from the mean of one observed sample.
Probability & statistics / English
Understanding Expectation of a Random Variable: Intuition for Expected Value and Linearity
Harvard Online · YouTube · 7:02
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Interpret expectation as a probability-weighted average for discrete and continuous variables, then use linearity without assuming independence.
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0:00A theoretical weighted average3:00Replace the sum by an integral6:00Linearity without independenceLearning script
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For an integrable discrete variable, expectation sums each possible value multiplied by its probability. This theoretical mean differs from the mean of one observed sample.
For an integrable variable with density f, expectation integrates over its support. The density supplies probability through areas rather than point probabilities.
When the expectations exist finitely, E(T₁+T₂)=E(T₁)+E(T₂), even if the two durations are dependent. Split a complicated total into simpler contributions.
Knowledge cards
Replace the sum by an integral
For an integrable variable with density f, expectation integrates over its support. The density supplies probability through areas rather than point probabilities.
Linearity without independence
When the expectations exist finitely, E(T₁+T₂)=E(T₁)+E(T₂), even if the two durations are dependent. Split a complicated total into simpler contributions.
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- Random variables ExplanationAt 0:00
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For an integrable discrete variable, expectation sums each possible value multiplied by its probability. This theoretical mean differs from the mean of one observed sample. For an integrable variable with density f, expectation integrates over its support. The density supplies probability through areas rather than point probabilities. When the expectations exist finitely, E(T₁+T₂)=E(T₁)+E(T₂), even if the two durations are dependent. Split a complicated total into simpler contributions.
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