9.1.4. Properties of Expectation

1. Linearity:

$E(aX + bY) = aE(X) + bE(Y)$

for constants $a, b$, and random variables $X, Y$.

The property of linearity in expectation states that if you have two random variables $X$ and $Y$, and you multiply them by constants $a$ and $b$ respectively, the expectation of their linear combination is equal to the weighted sum of their individual expectations. This means that you do not need to calculate the expectation of the entire expression at once; you can simply calculate the expectation of each variable, multiply by its constant, and sum the results.

Imagine you are measuring the height and weight of a group of people. If you were to create an index that combines both measurements by multiplying height by 2 and weight by 1.5, although you would need to consider the height and weight together, the average index can still be calculated by averaging the individual heights and weights first, then applying the multipliers and summing them. This property allows us to manage complex expressions by breaking them down.

2. Expectation of a Constant:

$E(c) = c$

The expectation of a constant value $c$ is simply the constant itself. This means that if you have a random variable that always takes the value $c$, the average or expected value is exactly $c$. There is no variability here since the value does not change; therefore, the mean is trivially the constant value.

Think about a scenario where you are promised a fixed amount of money, say $100, for performing a task—regardless of the task or any uncertainties involved. The expectation here is directly$100, as there are no other possible outcomes. This principle makes it easy to understand how fixed amounts behave in calculations concerning expectations.

3. Multiplicative Property (Independent Variables):

$E(XY) = E(X)E(Y)$ if $X$ and $Y$ are independent.

When you have two independent random variables $X$ and $Y$, the expectation of their product is equal to the product of their expectations. Independence here means that the outcome of one variable does not affect the outcome of the other. This property simplifies many calculations, especially in statistics and probability, as you can calculate the expectations separately and then multiply them, avoiding the complexities of calculating the expectation of the product directly.

Consider the number of heads you get from flipping two independent coins. The expected number of heads from flipping each coin is $0.5$. To find the expected number of heads from the two coins together, you simply multiply the expectations: $E( ext{Coin 1}) imes E( ext{Coin 2}) = 0.5 imes 0.5 = 0.25$. This shows that understanding the independence between events can significantly simplify the computation of expectations.

4. Expectation of Function of Random Variable:

$E[g(X)] = \sum g(x_i) p_i \quad \text{(discrete)}, \quad E[g(X)] = \int g(x) f(x) dx \quad \text{(continuous)}$

This property states that if you have a function $g(X)$ of a random variable $X$, the expectation of this function can be calculated either by summing over the values of the function weighted by their probabilities (for discrete random variables) or by integrating the function times the probability density function (for continuous random variables). This allows for flexibility in how you calculate expectations depending on the nature of the random variable.

Imagine you want to determine the expected value of the square of the number obtained when rolling a die. Here, your function $g(X)$ is the square of the outcome. You would first compute each square value (1, 4, 9, 16, 25, 36) and weigh these by the probability of rolling each number (1/6). This shows how expectations can be extended beyond just the original random variable.

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