Expected value is the weighted average of all possible values that a random variable can take on, with weights being their respective probabilities. It provides a measure of the center of the distribution of the variable.
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Expected value is often denoted by $E(X)$ or $\mu$.
In linear regression, expected value helps in predicting the mean response variable given certain predictor values.
For a discrete random variable, expected value is calculated as $E(X) = \sum{[x_i \cdot P(x_i)]}$ where $x_i$ are outcomes and $P(x_i)$ are their probabilities.
In continuous cases, it is computed using integrals: $E(X) = \int{x \cdot f(x)dx}$ where $f(x)$ is the probability density function.
Expected value plays a crucial role in determining the best-fit line in linear regression analysis.
Review Questions
What is the formula for calculating the expected value for a discrete random variable?
How does expected value assist in making predictions using a regression equation?
What symbol is commonly used to represent expected value?
Related terms
Random Variable: A variable whose possible values are numerical outcomes of a random phenomenon.
Probability Distribution: A function that describes the likelihood of obtaining possible values that a random variable can assume.
Linear Regression: A statistical method for modeling the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data.