Linearity of Expected Value: Suppose and are random variables and and are scalars. The following relationships hold:
Suppose for are independent identically distributed (iid) random variables. Then for and
is the stock price of AAPL at market close. is the sum of closing AAPL stock prices for 5 days. Then
Contrast this with the variance of . In other words, is a random variable that takes a value of 5 times the price of AAPL at the close of any given day. Then
The distinction between and is subtle but very important.
Variance of a Sample Mean:
In situations where the sample mean is a random variable over iid observations (i.e. the average price of AAPL over 5 days), the following formula applies: