expectation is calculated summing up the probability of an event multiplied by t

Question

expectation is calculated summing up the probability of an event multiplied by the value of the event. so to find the expectation of number of flips of a coin to get heads for the 1st time you would start by multiplying 1 flip by (1/2). then you would multiply 2 by (1/2)*(1/2). this 2nd probability is the probability of getting tails followed by heads. a) write the 1st 8 terms for the probability multiplied by value b) write the expectation as an infinite sum c) evalute the infinite sum d) does your answer make intuitive sense? explain

Explanation / Answer

In probability theory, the expected value (or expectation, mathematical expectation, EV, mean, or the first moment) of a random variable is the weighted average of all possible values that this random variable can take on. The weights used in computing this average correspond to the probabilities in case of a discrete random variable, or densities in case of a continuous random variable. From a rigorous theoretical standpoint, the expected value is the integral of the random variable with respect to its probability measure. The expected value may be intuitively understood by the law of large numbers: the expected value, when it exists, is almost surely the limit of the sample mean as sample size grows to infinity. More informally, it can be interpreted as the long-run average of the results of many independent repetitions of an experiment (e.g. a dice roll). The value may not be expected in the ordinary sense—the "expected value" itself may be unlikely or even impossible (such as having 2.5 children), just like the sample mean.

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