The expected mean (or expected value) of random variables $X$ and $Y$ are calculated using the following formulas:

1. For a single random variable $X$:
   $E(X) = \sum_{i} x_i P(X = x_i)$ if $X$ is discrete, or
   $E(X) = \int_{-\infty}^{\infty} x f_X(x) dx$ if $X$ is continuous, where $f_X(x)$ is the probability density function of $X$.

2. For two random variables $X$ and $Y$:

• If they are independent: $E(X + Y) = E(X) + E(Y)$
• If they have a joint distribution: $E(X + Y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} (x+y) f_{X,Y}(x, y) dx dy$ where $f_{X,Y}(x, y)$ is the joint probability density function.

Do you want more details or have any questions about this?

Here are 5 related questions you might find useful:

1. How do you calculate the expected value when $X$ and $Y$ are not independent?
2. What is the variance of the sum of $X$ and $Y$?
3. How does covariance affect the expected value of $X+Y$?
4. Can you explain the Law of the Unconscious Statistician?
5. How is the expected value used in real-world applications like finance or statistics?

Tip: Expected value is a measure of the center of a probability distribution, often interpreted as the long-run average outcome of a random variable.