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

• How do you calculate the expected value for a discrete random variable?
• Explain why expected value can be considered a weighted average.
• What property of expected value allows it to be expressed as $E[aX + b] = aE[X] + b$?

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.