# Provide an intuitive example of the linearity of expectation

Can anyone explain the linearity of expectation in an intuitive way? I have been trying to understand this for far too long now. Please don't use any equations and such, try to use real world examples or simple tools such as couples, red cards black cards etc.

For example, if there are 4 red cards and 5 black cards, the expectation of the amount of alternating cards would be equal to $$E[\text{alternating first pair} + \text{alternating second pair} + \dotsm + \text{alternating 8th pair}]$$, even though the first pair and second pair should be dependent. Why? Pleaes don't link to external articles and such, I really would like an intuitive explanation, and it seems like one hasn't been provided before.

I have a feeling linearity holds for this kind of example, because the next outcome is symmetric for each case, I.E if first card is black, second is red, the third is black, its symmetric in a way to see the first card is red, second is black and third is red, but I'm not sure how this relates to linearity.

• Linearity is a direct consequence of the basic properties of addition and multiplication of numbers. Any example of those properties would suffice, but is unlikely to be of any statistical interest!
– whuber
Mar 31 at 13:49
• I have a dice, the game is that I throw it... in expectation you will get $E[X] = 3.5$... now say that you win what you get and an additional 5 dollars, then $E[X+5] = E[X]+5=8.5$, because you can think "ok I will get 5 dollars... and then there is a stochastic game where there is some probability involved"... same holds for if I say that I'm going to give you double the amount you get... however, if you write the expectation as $E[f(x)] = \sum p(x)f(x)$, using the basic properties of summation, you can clearly see this Mar 31 at 13:54
• What are alternating cards?
– Eli
Mar 31 at 14:39
• For anyone trying to answer this question, I'd like to highlight that there are scenarios in which linearity of expectation is (IMO) genuinely not intuitive: math.stackexchange.com/a/2282639 I'd find it interesting if some answers shed light on scenarios like this (other than just by saying "it follows directly from the definition"). Mar 31 at 23:25
• Building on the comment of @helloworld , "expectation" is somewhat of a misnomer to the extent that it doesn't necessarily correspond to the (natural language meaning) "value that we would expect from the distribution". (That's why intro classes often also discuss the median and mode, because those are often better candidates for "value we'd expect".) I.e. "expectation" should be understood as a term of art, and intuition for it should be separate from intuition for "values that we'd expect". Apr 1 at 15:33

Linearity of expectation has everything to do with algebra. The concept is quite intuitive though because we often think in linear categories and we solve many linear equations in school. I am not sure if it is possible to answer your question without equations, but I'll try to make it intuitive though.

First of all, let's start with linearity. By "linear" we mean here the $$f(x) = a + b x$$ functions. To understand the following equations you only need to remember that the multiplication is distributive, so $$cx + cy = c(x+y)$$ and definitions of things like expected value, and joint distribution, etc.

## Multiplication by a constant

For a random variable $$X$$ and a constant $$c$$, the following holds

$$E[cX] = cE[X]$$

Example: According to some random source on the internet, the average height of women worldwide is 163 cm, this is the same as saying that the average height is 1.63 m, as 1 meter = 100 centimeters. Intuitively this is what we would expect, but to answer why does it hold, a little bit of algebra is needed.

First, recall that for a discrete random variable, the expected value is

$$E[X] = \sum_x x\, p(x)$$

then we have

\begin{align} E[cX] &= \sum_x c \, x \,p(x) \\ &= c \sum_x x \,p(x) \\ &= cE[X] \end{align}

$$E[X + c] = E[X] + c$$
Example: according to another random internet source, the average annual salary in the US is \$53,490. Now imagine that the US decides to introduce universal basic income and will give every citizen \$2,000 every month. How would the average income change? It would be \$53,490 + \$24,000 (12 months). It's also quite intuitive: every person would get \$24,000 extra money, so the total amount of money earned by US citizens would be their salaries + \$24,000 times the number of citizens. To get an average from it, divide the total by the number of citizens. Everyone got the same extra amount, so on average also everyone got that more money, hence by this amount the average income has changed.
\begin{align} E[X + c] &= \sum_x (x + c) \, p(x) \\ &= \sum_x x \,p(x) + \sum_x c \, p(x) \\ &= \sum_x x \,p(x) + c \sum_x p(x) & \text{move } c \text{ outside of the summation}\\ &= \sum_x x \,p(x) + c & \text{because by definition } \sum_x p(x) = 1\\ &= E[X] + c \end{align}