Question

Show that the expectation of the sum of two random variables defined over the same sample space is the sum of the expectations. Hint: Let \(p_{1}, p_{2}, \cdots, p_{n}\) be the probabilities associated with the \(n\) sample points; let \(x_{1}, x_{2}, \cdots, x_{n},\) and \(y_{1}, y_{2}\) \(\cdots, y_{n},\) be the values of the random variables \(x\) and \(y\) for the \(n\) sample points. Write out \(E(x), E(y),\) and \(E(x+y)\).

Short answer

The expectation of the sum of two random variables is the sum of their expectations: \(E(X + Y) = E(X) + E(Y)\).

Step-by-step solution

- Define the Expectation of a Single Random Variable

Let’s start by defining the expectation of a random variable. The expectation of a random variable, say \(X\), over a sample space with \(n\) sample points, where the probabilities associated with the sample points are \(p_1, p_2, \, \text{...}, \, p_n\), and the values of the random variable are \(x_1, x_2, \, \text{...}, \, x_n\) is given by:\[E(X) = \sum_{i=1}^{n} p_i x_i\]

- Define the Expectation for the Second Random Variable

Similarly, the expectation of another random variable, say \(Y\), with the same sample points and probabilities \(p_1, p_2, \, \text{...}, \, p_n\), and values \(y_1, y_2, \, \text{...}, \, y_n\) is given by:\[E(Y) = \sum_{i=1}^{n} p_i y_i\]

- Define the Expectation of the Sum of Two Random Variables

Now, consider the sum of these two random variables \(Z = X + Y\). The value of the random variable \(Z\) at each sample point is the sum of the values of \(X\) and \(Y\) at that point, so \(z_i = x_i + y_i\). Therefore, the expectation of \(Z\) is:\[E(Z) = E(X + Y) = \sum_{i=1}^{n} p_i z_i\]

- Substitute the Values

Substitute \(z_i\) with \(x_i + y_i\) in the expectation formula:\[E(X + Y) = \sum_{i=1}^{n} p_i (x_i + y_i)\]

- Expand the Sum

Distribute the probability \(p_i\) through the sum to get:\[E(X + Y) = \sum_{i=1}^{n} p_i x_i + \sum_{i=1}^{n} p_i y_i\]

- Recognize Individual Expectations

Notice that the first sum is the expectation of \(X\) and the second sum is the expectation of \(Y\). Thus, we have:\[E(X + Y) = E(X) + E(Y)\]

Key concepts

sum of random variables

When adding two random variables, we are interested in understanding the behavior and properties of their combined outcome. Consider two random variables, say, X and Y. Suppose they are defined over the same sample space with corresponding values at each sample point.

The sum of these two variables, denoted as Z = X + Y, means that at each sample point, the value of Z is simply the sum of the values of X and Y. For example, if at a certain sample point X = 3 and Y = 4, then Z = X + Y = 7 at that point.

In general terms, if we look at multiple sample points, the value of Z at each point will be the combined outcome of the values of X and Y at those respective points. This principle is crucial as it lays the groundwork for understanding how various statistical measures, like expectation, behave when dealing with the sum of random variables.

probability

Probability is the measure of the likelihood that an event will occur. In the realm of random variables, probabilities are assigned to the different possible outcomes in a sample space. Each sample point in our sample space has a probability associated with it.

Let's say our sample space has n points, and the probabilities associated with these points are denoted as p_1, p_2, ..., p_n. The sum of these probabilities is always 1, as one of the outcomes must happen.

When dealing with random variables, these probabilities help us in calculating the expectation of the random variables. For instance, if X and Y are random variables with values x_1, x_2, ..., x_n and y_1, y_2, ..., y_n, respectively, at these sample points, the probabilities, p_1, p_2, ..., p_n, are used to calculate the expectations E(X) and E(Y).

expectation

The expectation (or expected value) of a random variable gives us a measure of the central tendency of its distribution. It represents the average outcome if the experiment from which the random variable arises is repeated many times.

Mathematically, the expectation of a random variable X, denoted E(X), is calculated as: \[E(X) = \sum_{i=1}^{n} p_i x_i\] Here, p_i is the probability of the i-th outcome, and x_i is the value of the random variable X at that outcome.

Similarly, if we have another random variable Y over the same sample space, its expectation E(Y) is given by: \[E(Y) = \sum_{i=1}^{n} p_i y_i\]Now, according to the linearity property of expectations, the expectation of the sum of the two random variables, X + Y, is the sum of their expectations: \[E(X + Y) = E(X) + E(Y)\]This relationship holds because when we calculate E(X + Y), we distribute the probabilities over the sum, which allows us to break it down into the sum of individual expectations.