Discrete random expectation transformation

For a discrete random variable, called X, if we know that there is another DRV which we can write as Y=aX+b - where a and b are constants:

E(Y)=a\,E(X)+b

Explanation

If we add b to the result of any outcome of the DRV, we would expect to get, on average, a value b more than before (because we’ve shifted our entire distribution by b).
When multiplying by a, we multiply the random variable X by a, so the average we can expect to get from X (so E(X)) is a times what it was.

flashcards

Question	Answer
E(Y) = aE(X) + b when Y = aX + b	For a discrete random variable X, if Y = aX + b with constants a and b, then the expected value of Y is a times the expected value of X plus b.
Why does adding b increase the expectation by b?	Adding b to every outcome shifts the entire distribution upward by b, so the average (expected value) also increases by b.
Why does multiplying by a multiply the expectation by a?	Multiplying the random variable X by a scales all outcomes by a, so the average (expected value) becomes a times the original average E(X).

Inlinks

transforming discrete random probabilities

Outlinks

discrete random variable random variable flashcards