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.
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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.
Expected value can be thought of as a weighted average, where each outcome is weighted by its probability.
If a random variable X has values $x_1, x_2, ..., x_n$ with corresponding probabilities $p_1, p_2,..., p_n$, then $E[X] = x_1p_1 + x_2p_2 + ... + x_np_n$.
Expected value is linear: $E[aX + b] = aE[X] + b$, where 'a' and 'b' are constants.
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.