7 11: Expected Value Mathematics LibreTexts

For continuous random variables, the calculation involves integration over the entire probability density function. Expected value is very common in making insurance decisions. Insurance companies take into account various factors to determine probabilities that policy holders will have an accident, fire, or death. They set prices on policies to make ensure the company will remain profitable even though they will occasionally pay out large monetary claims. In other words, even though the insurance companies expect some negative outcomes, it is important to their bottom line that their expected return remain positive. Since the expected value is not zero this is not a fair game.

For example, while playing poker, or maybe analyzing a lottery system. This concept is also used in the field of Artificial Intelligence(AI) a lot to understand real-life scenarios and actions. While the expected value provides a mathematical average of possible outcomes, it does not guarantee specific results in individual cases. Variability and volatility in real-world scenarios mean that actual outcomes can significantly deviate from the expected value. Hence, it’s also essential to consider the distribution and risk of outcomes, not just the average. You may also hear the term “the expectation of X” which is a short way of saying “the expected value of the random variable X”.

How to calculate the expected value?

The average value of the number of successes get closer to the EV of the number of successes as we increase the number of trials. The average value converges (get closer) to the EV as we increase the number of trials. When we select a sample, we use the sample mean to estimate the population mean or the expected value. Players can bet that the next roll of the dice will be “any craps” (a sum of 2, 3, or 12). Players can bet on the next roll of the dice being a sum of 7.

Not all problems involving expectation are about games or money. The outcome can be anything that is numerical as the following problem shows. Expected Value \((EV)\) is the average gain or loss if an experiment or procedure with a numerical outcome is repeated many times. The expected value, also known as the mean value, is the total of all potential decision outcomes multiplied by each possibility’s probability.

Most elementary courses do not cover the geometric, hypergeometric, and Poisson. Your instructor will let you know if he or she wishes to cover these distributions. To get the standard deviation σ, we simply take the square root of variance σ2. Therefore, we expect a newborn to wake its mother after midnight 2.1 times per week, on the average. The men’s soccer team would, on the average, expect to play soccer 1.1 days per week.

The values are expressed as the variable x with a numerical subscript. The subscript indicates the index of the element in the sequence. There are many areas in which expected value is applied and it’s difficult to give a comprehensive list. It is used in a variety of calculations by natural scientists, data scientists, statisticians, investors, economists, financial institutions, and professional gamblers, to name just a few. A stronger linearity property holds, which involves two (or more) random variables.

Many conflicting proposals and solutions had been suggested over the years when it was posed to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré in 1654. Méré claimed that this problem could not be solved and that it showed just how flawed mathematics was when it came to its application to the real world. Pascal, being a mathematician, decided to work on a solution to the problem. Let X be the random variable which represents profit when the “04” ticket was drawn.

This distribution is frequently used in scenarios where you’re interested in knowing how soon or how late the first success will occur. To get the fourth column xP(x) in the table, we simply multiply the value x with the corresponding probability P(x). If calculated expected number is negative, it means you are likely get an average loss, which is unfavorable. Expected value calculation helps us to predict the expected average when a risk is taken with an uncertain outcome. Let be a matrix with random entries, such that all its entries have expected value equal to .

We see a nearly equal number of data points on either side of the vertical line, so the expected value or the mean gives us a measure of the data center. Multiply each outcome by its probability and sum to get the expected value. If we have sample data, we use the sample mean to estimate the population mean or expected value. Whether we increase the number of trials (samples) or the number of persons within each sample, the average will get closer to the EV for the average. Whether we increase the number of trials or the number of rollings within each trial, the average will get closer to the EV for the average.

• Expected value, often abbreviated as EV, is a statistical concept that calculates the average outcome of a random variable over a large number of trials.
• The expected value is a way to summarize all this information in a single numerical value.
• The expected value is a theoretical concept that represents the long-run average outcome if an experiment were repeated many times.
• Data can be retrieved using basic algebraic manipulations and expressions for calculating the expected value.

Random Variables and Expectations

Well, now you have the formal definition of expected value. In the next sections, I’m going to give you some intuition about what exactly it measures and clear up some potential confusions. You do this by multiplying each possible value by its respective probability and add the products. In fact, let’s start with our very first example and finally answer the question from the end of the previous section. I’m going to show you how to calculate the expected value in this particular example without too much explanation and I’ll give the bc game official details afterwards. A carnival game consists of drawing one ball from a box containing two yellow balls, five red balls, and eight white balls.

We then add all the products in the third column to get the mean/expected value of X. However, Jensen’s inequality tells us thatif is convex and if is concave. This rule is again a consequence of the fact that the expected value is a Riemann-Stieltjes integral and the latter is linear. This property has been discussed in the lecture on the Expected value. It can be proved in several different ways, for example, by using the transformation theorem or the linearity of the Riemann-Stieltjes integral. We see that males have longer heights (histogram shifted to the right), so males have a higher expected value for the average height.

We do many random processes that generate these random variables to get the EV or the mean. You are about to play a game in which you draw 3 ping-pong balls without replacement from a barrel. A life insurance actuary estimates the probabilities of \(X\), a person’s life expectancy, with the probability density function as described above. Let \(X_n\) be the random variable that represents the number of rolls required to get \(n\) consecutive sixes. The first theorem shows that translating all variables by a constant also translates the expected value by the same constant.

In statistics, where one seeks estimates for unknown parameters based on available data gained from samples, the sample mean serves as an estimate for the expectation, and is itself a random variable. This division is the only equitable one when all strange circumstances are eliminated; because an equal degree of probability gives an equal right for the sum hoped for. In Dutch mathematician Christiaan Huygens’ book, he considered the problem of points, and presented a solution based on the same principle as the solutions of Pascal and Fermat.

A game that has an expected value of zero is called a fair game. Where each \(X\) is the net amount gained or lost on each outcome of the experiment and \(P\) is the probability of that outcome. As in the procedure above, we compute the expected value by multiplying the amount gained/lost for each outcome by the probability of that outcome. If exists and is finite, we say that is an integrable random variable, or just that is integrable. An important property of the expected value, known as transformation theorem, allows us to easily compute the expected value of a function of a random variable. A completely general and rigorous definition of expected value is based on the Lebesgue integral.

For more details on this, check out my posts on probability distributions and the mean and variance of probability distributions. As discussed above, there are several context-dependent ways of defining the expected value. The simplest and original definition deals with the case of finitely many possible outcomes, such as in the flip of a coin. With the theory of infinite series, this can be extended to the case of countably many possible outcomes. It is also very common to consider the distinct case of random variables dictated by (piecewise-)continuous probability density functions, as these arise in many natural contexts. In probability theory, the expected value (often denoted as $EX$ for a random variable $X$) represents the average or mean value of a random experiment if it were repeated many times.

For example, the expected value when rolling a fair six-sided die is 3.5, even though it’s impossible to actually roll a 3.5. The expected value represents a weighted average, not necessarily a possible outcome. The expected value is a theoretical concept that represents the long-run average outcome if an experiment were repeated many times. The average (or sample mean) is calculated from observed data.

Conditional expected value is an extension of the expected value concept, applied when the outcome of a random variable depends on a certain condition being met. It reflects the average outcome considering that a specific event has already occurred. In the next example, we will demonstrate how to find the expected value and standard deviation of a discrete probability distribution by using relative frequency. The expected value calculator helps you to calculate expected value, which is the average prediction of a probability distribution. The expected value of is a weighted average of the values that can take on. Formally, the expected value is the Lebesgue integral of , and can be approximated to any degree of accuracy by positive simple random variables whose Lebesgue integral is positive.

Since the partitioning of \(\Omega\) has many different representations, we must show that, given any choice of representation, we still wind up with a well-defined notion of expectation. Models the number of trials needed to get the first success in a series of independent trials. Used for scenarios like the number of coin flips until the first heads. Models the number of successes in a fixed number of independent trials, each with the same probability of success. Used for scenarios like coin flips, yes/no surveys, or pass/fail tests.

As we noted, the expected value of an experiment is the mean of the values we would observe if we repeated the experiment a large number of times. (This interpretation is due to an important theorem in the theory of probability called the Law of Large Numbers.) Let’s use that to interpret the results of the previous example. The Las Vegas casino Magnicifecto was having difficulties attracting its hotel guests down to the casino floor. The empty casino prompted management to take drastic measures, and they decided to forgo the house cut. They decided to offer an “even value” game–whatever bet size the player places \((\)say \($A),\) there is a 50% chance that he will get \( +$A \), and a 50% chance that he will get \( – $A \). They felt that since the expected value of every game is 0, they should not be making or losing money in the long run.

Roughly speaking, this integral is the limiting case of the formula for the expected value of a discrete random variable Here replaces (the probability of ) and the integral sign replaces the summation sign . Think of expected value as the “fair price” or “average outcome” you would expect in the long run if you were to repeat an experiment or process many times. It’s a crucial tool in decision-making, risk assessment, and understanding random phenomena. Our Expected Value Calculator simplifies the process of calculating expected values for any probability distribution. Whether you’re a student learning statistics or a professional analyzing data, this comprehensive guide will show you exactly how to find expected value using our calculator and manual methods.