What Expectation is
Expectation is a concept used in probability theory and statistics to represent the value of a random variable. It is the average value of the random variable’s possible outcomes.
Steps for Expectation:
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Identify the random variable: The first step in calculating expectation is to identify the random variable. A random variable is a variable whose possible values are numerical outcomes of a random phenomenon.
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Find the probability of each outcome: Once the random variable has been identified, the probabilities of each of its possible outcomes must be found.
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Multiply the probability of each outcome by its value: For each outcome, multiply its probability by its value.
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Sum the products: Sum the products of each probability-value pair to get the expectation.
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Simplify the expression, if necessary: If the expression contains multiple terms, simplify it using the rules of algebra.
Examples
- Expectation of a coin flip is 0.5, which means that the probability of getting either heads or tails is equally likely.
- Expectation of a dice roll is 3.5, which means that the average of all possible outcomes is 3.5.
- Expectation of a normal distribution is the mean, which is the sum of all possible values divided by the total number of values.
- Expectation of the mean of a sample population is equal to the population mean.
- Expectation of the variance of a sample population is equal to the population variance divided by the sample size.