Expected value is a statistical concept that represents the average or central tendency of a probability distribution. It is the sum of the products of each possible outcome and its corresponding probability, providing a measure of the typical or expected outcome in a probabilistic scenario.
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The expected value of a discrete random variable is calculated by multiplying each possible outcome by its corresponding probability and then summing these products.
Expected value can be used to make decisions under uncertainty by providing a measure of the typical or average outcome that can be expected.
The expected value is a linear function, meaning that the expected value of a sum of random variables is the sum of their individual expected values.
In the context of probability, expected value represents the long-term average or mean of a probability distribution if the experiment is repeated many times.
Expected value is a fundamental concept in decision theory, game theory, and various other areas of mathematics and statistics.
Review Questions
Explain how the expected value is calculated for a discrete random variable.
For a discrete random variable, the expected value is calculated by multiplying each possible outcome by its corresponding probability and then summing these products. Mathematically, the expected value of a discrete random variable $X$ is expressed as $E[X] = \sum_{i=1}^n x_i \cdot P(X=x_i)$, where $x_i$ represents the possible outcomes and $P(X=x_i)$ is the probability of each outcome occurring.
Describe how the expected value can be used to make decisions under uncertainty.
The expected value provides a measure of the typical or average outcome in a probabilistic scenario, which can be used to make informed decisions under uncertainty. By calculating the expected value of different alternatives or outcomes, decision-makers can compare the potential payoffs and choose the option with the highest expected value. This allows for a more rational and informed decision-making process, especially when dealing with situations involving risk or multiple possible outcomes.
Analyze how the linearity property of expected value can be applied in probability and statistics.
The linearity property of expected value states that the expected value of a sum of random variables is equal to the sum of their individual expected values. This property is particularly useful in probability and statistics, as it allows for the calculation of expected values of more complex random variables or scenarios. For example, in the context of a portfolio of investments, the expected return of the portfolio can be calculated as the weighted average of the expected returns of the individual investments. This linearity property simplifies the analysis and decision-making process in many probabilistic and statistical applications.
Related terms
Probability Distribution: A probability distribution is a function that assigns probabilities to the possible outcomes of a random variable, describing the likelihood of each outcome occurring.
Random Variable: A random variable is a variable that can take on different values, each with a certain probability of occurring, in a probabilistic or statistical experiment.
Variance: Variance is a measure of the spread or dispersion of a probability distribution, indicating how much the values tend to deviate from the expected value.