Conditional expectation is a statistical concept that measures the expected value of a random variable given the knowledge of another random variable. It represents the average outcome of a random variable, taking into account the information or conditions provided by another random variable.
The formula for conditional expectation is E(X|Y) = ∑x P(X=x|Y) * x, where E(X|Y) represents the conditional expectation of X given Y, P(X=x|Y) is the probability of X taking on a specific value x given Y, and x is the specific value of X.
Unconditional expectation is the average value of a random variable without taking into account any conditions or additional information. Conditional expectation, on the other hand, takes into account the information provided by another random variable and calculates the average value of the first random variable based on that information.
Conditional expectation has various applications in fields such as economics, finance, and biology. For example, in economics, it is used to predict consumer behavior based on income levels. In finance, it is used to estimate stock prices based on market conditions. In biology, it is used to predict the survival rate of a species based on environmental factors.
Conditional expectation and conditional probability are closely related concepts. Both involve calculating the likelihood of an event based on certain conditions or information. However, while conditional probability measures the likelihood of an event occurring given some condition, conditional expectation measures the expected value of a random variable given a specific condition or information.