Expected value

Definition

Expected value is the long-term average or mean of a random variable, calculated as the sum of all possible values each multiplied by their probability of occurrence. It provides a measure of the center of the distribution for a discrete random variable.

5 Must Know Facts For Your Next Test

1. The formula for expected value (E[X]) is $E[X] = \sum{[x_i \cdot P(x_i)]}$ where $x_i$ are the values and $P(x_i)$ are their probabilities.
2. Expected value can be thought of as a weighted average, where each outcome is weighted by its probability.
3. If a random variable X has values $x_1, x_2, ..., x_n$ with corresponding probabilities $p_1, p_2,..., p_n$, then $E[X] = x_1p_1 + x_2p_2 + ... + x_np_n$.
4. Expected value is linear: $E[aX + b] = aE[X] + b$, where 'a' and 'b' are constants.
5. The concept of expected value is foundational in decision theory and economics, helping to assess risks and benefits quantitatively.

Review Questions

Related terms

Discrete Random Variable:

A type of random variable that can take on only specific, distinct values.

Probability Mass Function (PMF): A function that gives the probability that a discrete random variable is exactly equal to some value.

Standard Deviation:

A measure of the amount of variation or dispersion in a set of values.