Förväntningsvärden

Förväntningsvärden, also known as expected values, are a fundamental concept in probability theory and statistics. They represent the long-term average value of a random variable. In other words, if an experiment is repeated many times, the expected value is the average of the outcomes. It is calculated by multiplying each possible outcome by its probability and then summing these products.

The expected value of a discrete random variable X, denoted as E(X), is given by the formula:

E(X) = Σ [x_i * P(X = x_i)]

where x_i are the possible values of X and P(X = x_i) is the probability of each value.

For a continuous random variable, the expected value is calculated using integration. If X is a continuous random variable with probability density function f(x), then the expected value E(X) is given by:

E(X) = ∫ [x * f(x)] dx

Expected values have several important properties. For example, the expected value of the sum of two random variables is the sum of their expected values, provided the expected values exist. This property is known as linearity of expectation.

Expected values are widely used in various fields, including finance, insurance, and operations research. They provide a way to quantify uncertainty and make informed decisions under uncertainty. For instance, in finance, the expected value of a stock's return is used to evaluate the stock's potential profitability.