Expected value what is

Expected Value In a probability distribution , the weighted average of possible values of a random variable, with weights given by their respective theoretical probabilities, is known as the expected value , usually represented by E x. Subjects Near Me. Download our free learning tools apps and test prep books. Varsity Tutors does not have affiliation with universities mentioned on its website.

Plus, the next outcome is two and has a probability of 0. Plus, the outcome three has a probability of 0. And then last but not least, we have the outcome four workouts in a week, that has a probability of 0. Well, we can simplify this a little bit.

Zero times anything is just zero. So, one times 0. Two times 0. Three times 0. And then four times. And so, we just have to add up these numbers. So, we get 0. Let's add 'em all together. And so, let's see, five plus five is And then this is two plus eight is 10, plus seven is 17, plus four is Because of the law of large numbers , the average value of the variable converges to the EV as the number of repetitions approaches infinity. The EV is also known as expectation, the mean or the first moment.

EV can be calculated for single discrete variables, single continuous variables, multiple discrete variables, and multiple continuous variables. For continuous variable situations, integrals must be used. To calculate the EV for a single discrete random variable, you must multiply the value of the variable by the probability of that value occurring.

Take, for example, a normal six-sided die. Once you roll the die, it has an equal one-sixth chance of landing on one, two, three, four, five, or six. Given this information, the calculation is straightforward:. If you were to roll a six-sided die an infinite amount of times, you see the average value equals 3. Tools for Fundamental Analysis. Portfolio Management.

Financial Ratios. Financial Analysis. Investing Essentials. Your Privacy Rights. In other words, each possible value the random variable can assume is multiplied by its assigned weight, and the resulting products are then added together to find the expected value.

The weights used in computing this average are the probabilities in the case of a discrete random variable that is, a random variable that can only take on a finite number of values, such as a roll of a pair of dice , or the values of a probability density function in the case of a continuous random variable that is, a random variable that can assume a theoretically infinite number of values, such as the height of a person. From a rigorous theoretical standpoint, the expected value of a continuous variable is the integral of the random variable with respect to its probability measure.

Since probability can never be negative although it can be zero , one can intuitively understand this as the area under the curve of the graph of the values of a random variable multiplied by the probability of that value. Thus, for a continuous random variable the expected value is the limit of the weighted sum, i. Without too much effort, you can compute the following probabilities:.

This calculation can be easily generalized to more complicated situations. The formula for the bonus is:. We could have calculated the same value by taking the expected number of children and plugging it into the equation:.

The intuitive explanation of the expected value above is a consequence of the law of large numbers: the expected value, when it exists, is almost surely the limit of the sample mean as the 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. To empirically estimate the expected value of a random variable, one repeatedly measures observations of the variable and computes the arithmetic mean of the results.

If the expected value exists, this procedure estimates the true expected value in an unbiased manner and has the property of minimizing the sum of the squares of the residuals the sum of the squared differences between the observations and the estimate. The law of large numbers demonstrates under fairly mild conditions that, as the size of the sample gets larger, the variance of this estimate gets smaller. This property is often exploited in a wide variety of applications, including general problems of statistical estimation and machine learning, to estimate probabilistic quantities of interest via Monte Carlo methods.