Expected Value Calculator

Given a discrete random variable $$X$$ that takes values $$x_1, x_2, \ldots, x_n$$ with corresponding probabilities $$p_1, p_2, \ldots, p_n$$, the expected value is:

$$E(X) = \sum_{i=1}^{n} x_i \cdot p_i$$

The variance measures the spread of the distribution around the mean:

$$\text{Var}(X) = \sum_{i=1}^{n} p_i \cdot (x_i - E(X))^2$$

And the standard deviation is simply the square root of the variance:

$$\sigma(X) = \sqrt{\text{Var}(X)}$$

Important requirement: The probabilities should sum to 1 (i.e., $$\sum p_i = 1$$). If they do not, the results will still be computed but may not represent a valid probability distribution.

The expected value is a weighted average where each outcome is weighted by how likely it is to occur. It does not need to be a value that the random variable can actually take — for instance, the expected value of a fair six-sided die is 3.5, which is not a face value.

Variance quantifies how much the outcomes deviate from the expected value on average (in squared units), while the standard deviation brings this back to the original units for easier interpretation.

The expected value represents the theoretical average outcome over infinitely many trials. If E(X) = 25, then repeating the random process many times would yield an average close to 25.

The variance indicates how spread out the outcomes are. A variance of 0 means all probability is concentrated on a single value. Larger variance means more uncertainty.

The standard deviation provides the spread in the same units as the original values, making it more intuitive than variance for interpretation.