LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
The expected value of a discrete random variable is the sum of each outcome times its probability. It is not the outcome you see most often. A fair die has six faces at probability 1/6 each, and that sum is 3.5, which is not a face you can roll. Variance is the expected squared distance from that mean. A shift moves the mean and leaves the variance unchanged. A scale factor multiplies the variance by the square of the factor. Correlation takes covariance, which has awkward units, and rescales it to sit between negative 1 and 1.
Why it matters
People use “expected” to mean “likely.” The die is the counterexample. No face is likelier than another, and the number the definition returns cannot occur. The definition is still the right object for a repeated game: if a roll pays its face in dollars, 3.5 is the long-run average payment per roll. A rolled 4 is already above that average. A rolled 2 is below it. That is a decision about one more roll, not a claim that 3.5 will appear.
The idea in plain words
List the die faces 1 through 6. Factor out the shared 1/6 and add the faces. Divide by 6 and you get 3.5. NumPy’s mean of that same list is 3.5, because that is what the mean is doing.
A shift of t adds t to every face and therefore adds t to the mean. The lecture checks a shift of 1, which moves 3.5 to 4.5, and a shift of 5, which moves it to 8.5. A scale of r multiplies every face and therefore multiplies the mean by r. A scale of 2 turns 3.5 into 7. Do the scale first and the shift second. Scaling 3.5 by 2 and then adding 5 gives 12, which is the lecture’s combined check. The shift is not multiplied by r. Linearity is this pair of facts. It also applies to a sum of random variables whether or not they are independent. Products are the case the lecture tells you not to distribute blindly.
Variance in the form they compute is the mean of the squared values, minus the square of the mean. On the die that difference is about 2.917. The picture they prefer is a cloud of points and a horizontal line at the mean. Shift every point up by the same amount and the line moves with them. Distances to the line do not change, so the variance does not change. Multiply every point by r and the distances stretch. Because variance squares those distances, r comes out as r squared. A scale of 2 multiplies the die’s variance by 4. A scale of 1 leaves it alone. Adding 100, or adding 0, does nothing to it.
They illustrate the square with a 252-day count. If a return series is multiplied by the square root of 252, the variance is multiplied by 252. That is the same algebra as the die. They also say this annualizing sketch was improvised and deserved another look. Keep the algebra. Do not treat the sketch as a finished market convention.
How to use it
Compute a mean from outcomes and probabilities, or from a sample list, and only then talk about “what you expect.” If the mean is not a possible outcome, say so, as with 3.5. Apply shifts and scales to the mean with linearity, and apply them to the variance with the square rule, not with linearity.
What usually goes wrong
Calling 3.5 the most common roll is the definition error. Every face has probability 1/6. The mode is not unique, and the mean is not a mode.
Questions people ask
Why is the expected roll 3.5 if a die cannot show 3.5?
Because expectation adds face times probability. It is not required to be an attainable face. For a fair die the sum is 3.5, and that is the average payment if each face pays its number.
What does a shift of 5 do to the die?
It moves the mean from 3.5 to 8.5 and leaves the variance at about 2.917. A scale of 2 moves the mean to 7 and multiplies the variance by 4. Scale by 2 and then shift by 5 and the mean is 12.
Does zero correlation mean independence?
No. Independence implies covariance zero, in the setting the lecture uses, and covariance zero implies uncorrelated. The reverse is false. Their independent normal and Poisson draws were a case where the variances nearly added. The addition did not prove the independence.