# Expected Value Computation

Compute the sum of the arithmetic series 2 + 5 + 8 + 11 + 14 Calculate the expected value E(X), the variance σ2 = Var(X), and the standard. What is the proper way to compute effectively (fast) the expected value E(x) in a case when I have approximation of probability desity function f(x) by probability. In many cases in practice, it is necessary to specify only the expectation value and standard deviation of each PDF, i.e. the best estimate of each quantity [ ]. In both US GAAP (computation of provision amount applying figures based on past experience) and IFRS (IAS ) the expected value method is used, i.e. In order to check for convexity, first and second derivatives of VaR are calculated. The same calculations are then repeated for expected shortfall, which is often. select sum([Probability]*[Value]) ExpectedValue from CRM. image. While this approach works well with a large number of leads of similar size, for. In many cases in practice, it is necessary to specify only the expectation value and standard deviation of each PDF, i.e. the best estimate of each quantity [ ]. Compute the expected value and the standard deviation of X, (1) without (2) with considering Compute the expected value for the prize of such a scratch card.

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## Expected Value Computation Video

How To Calculate Expected Value

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If the well logs are unsatisfactory, an abandonment cost of 40, dollars will be incurred at year 1. The above decision making process can be displayed in the following figure.

These types of graphs are called decision trees and are very useful for risk involved decisions. Each circle indicates a chance or probability node, which is the point at which situations deviate from one another.

Costs are shown in thousands of dollars. The main body of the tree starts from the first node on the left with a time zero lease cost of , dollars that is common between all four situations.

The next node, moving to the right, is the node that includes a common drilling cost of , dollars. At this node, an unsatisfactory and abandonment situation with a cost of 40, dollars in the first year situation D is deviated from other situations a branch for situation D is deviated from tree main body.

The next node on the right third node is the node where situation A, B, and C three separate branches get separated from each other.

In the beginning of each branch is the probability of that situation, and in the end of it, amounts due to that situation including cost, income, and salvage value are displayed.

So, there are four stations: Situation A: Successful development that yields the income of dollars per year Situation B: Successful development that yields the income of dollars per year Situation C: Failure that yields salvage value of dollars in the end of year two Situation D: Failure that yields abandonment cost of 40 dollars in the end of year one.

So, first we need to calculate ENPV for each situation:. Project ENPV is slightly less than zero compared to the total project cost of 1 million dollars, therefore, slightly unsatisfactory or breakeven economics are indicated.

That will be paid for all the cases. Again, this cost is paid for all the cases. And we need to close the wells and pay the abandonment cost and so on.

In this case, we will face three cases. So we can summarize the information here. So decision tree is a very helpful graph that can help us separate the possible cases here.

So I will explain this in this graph. So we start from the left hand side, initial investment for the lease at the present time.

We write the cost or income here. And in front of that we write the probability. This 1 plus is to show that this is the same year as this year.

These are happening in the same year. But because these cases are deviated from the main branch, we draw another branch for these, to separate these from the main branch.

And we will have three new cases in the after. So years are here. So every value under the same column has the same year dimension. So as we can see here, we have four main cases here.

Case A, case B, case C, and case D. So the first step to approach this problem and calculate the expected NPV is to calculate the probability of each case.

So in order to calculate the probabilities of each case, we go back to the decision tree. We start from the right hand side for each case.

For example, for case A. So I start from the right hand side. For example, case A, I start moving from right hand side toward the left.

And go to the main branch. So I will multiply. I start moving from right hand side along each branch to the left, and I multiply the probabilities that I see on the way.

In the second step, I draw the timeline and I separate the cases from each other. The first row I write the probabilities. You remember, this was in case the well logs look good.

This was happening at year 1. The lease cost is at year 0. And the income from year 2 to year Case 3. And the income for other years is going to be 0.

So in case of case D, which I call it failure case. I pay the lease cost at present time. I pay the drilling cost at year 1.

But the well logs are not looking good enough to pay the completion costs. So now that I have this table calculating the expected NPV.

And I make a summation over all that. So here, you can see this is the equation for case A to calculate the NPV. This is the lease cost at present time.

It doesn't need to be discounted. And I need to discount this for one year because they start from year 2. Please note that the salvage is happening at year 2 for case C, so I need to discuss that for two years.

And I write the NPV for each case in the last column. I multiplied probability by the NPV for each case. And I wrote that too in this column. And the summation of all these values here is going to give me the expected NPV for this project.

And the conclusion would be because expected NPV is slightly negative, is slightly less than 0. We can conclude that this project is not very economically satisfactory.

However, convergence issues associated with the infinite sum necessitate a more careful definition. A rigorous definition first defines expectation of a non-negative random variable, and then adapts it to general random variables.

Unlike the finite case, the expectation here can be equal to infinity, if the infinite sum above increases without bound. By definition,.

A random variable that has the Cauchy distribution  has a density function, but the expected value is undefined since the distribution has large "tails".

The basic properties below and their names in bold replicate or follow immediately from those of Lebesgue integral. Note that the letters "a.

We have. Changing summation order, from row-by-row to column-by-column, gives us. The expectation of a random variable plays an important role in a variety of contexts.

For example, in decision theory , an agent making an optimal choice in the context of incomplete information is often assumed to maximize the expected value of their utility function.

For a different example, in statistics , where one seeks estimates for unknown parameters based on available data, the estimate itself is a random variable.

In such settings, a desirable criterion for a "good" estimator is that it is unbiased ; that is, the expected value of the estimate is equal to the true value of the underlying parameter.

It is possible to construct an expected value equal to the probability of an event, by taking the expectation of an indicator function that is one if the event has occurred and zero otherwise.

This relationship can be used to translate properties of expected values into properties of probabilities, e. The moments of some random variables can be used to specify their distributions, via their moment generating functions.

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 , since most quantities of interest can be written in terms of expectation, e.

In classical mechanics , the center of mass is an analogous concept to expectation. For example, suppose X is a discrete random variable with values x i and corresponding probabilities p i.

Now consider a weightless rod on which are placed weights, at locations x i along the rod and having masses p i whose sum is one.

The point at which the rod balances is E[ X ]. Expected values can also be used to compute the variance , by means of the computational formula for the variance.

A very important application of the expectation value is in the field of quantum mechanics. Thus, one cannot interchange limits and expectation, without additional conditions on the random variables.

A number of convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below.

There are a number of inequalities involving the expected values of functions of random variables. The following list includes some of the more basic ones.

For other uses, see Expected value disambiguation. Retrieved Wiley Series in Probability and Statistics. The American Mathematical Monthly.

English Translation" PDF. A philosophical essay on probabilities. Dover Publications. Fifth edition. select sum([Probability]*[Value]) ExpectedValue from CRM. image. While this approach works well with a large number of leads of similar size, for. Compute the expected value and the standard deviation of X, (1) without (2) with considering Compute the expected value for the prize of such a scratch card. Many translated example sentences containing "estimate of the expected" market-conform: the highest estimate of the expected value for paper [ ] ordinary​. cumulant generating function. CumulativeDistributionFunction. cumulative distribution function. Decile. deciles. ExpectedValue. compute expected values. Calculate Market Value at Risk (VaR) and Expected Shortfall using Variance Covariance Method (VCM) based on the chosen confidence level and holding.

## Expected Value Computation Video

How to find an Expected Value Use of methods other than the ones used to establish the expected values may give different values Handy Aufbauspiele the ones indicated. The constructors are used to create objects that wrap distributions. And in fact, we could win or loose this lead which means a Casino Deutschland 18 of 0 or 1 million but nothing in between. Can you post your data? Cancel Copy to Clipboard. This question helps us to combat spam. The method of Claim 1, wherein calculation of attenuation is performed by comparing measured values to expected values from a model medium calculated using Monte Carlo techniques. Hilfe zum Textformat. Find the sum of the products. The expected value EV of a set of outcomes is the sum of the individual products of the value times its probability.

Using whatever chart or table you have created to this point, add up the products, and the result will be the expected value for the problem.

Interpret the result. The EV applies best when you will be performing the described test or experiment over many, many times.

For example, EV applies well to gambling situations to describe expected results for thousands of gamblers per day, repeated day after day after day.

However, the EV does not very accurately predict one particular outcome on one specific test. Over many many draws, the theoretical value to expect is 6.

But if you were gambling, you would expect to draw a card higher than 6 more often than not. Method 2 of Define all possible outcomes.

Calculating EV is a very useful tool in investments and stock market predictions. As with any EV problem, you must begin by defining all possible outcomes.

Generally, real world situations are not as easily definable as something like rolling dice or drawing cards. For that reason, analysts will create models that approximate stock market situations and use those models for their predictions.

These results are: 1. Earn an amount equal to your investment 2. Earn back half your investment 3. Neither gain nor lose 4. Lose your entire investment.

Assign values to each possible outcome. In some cases, you may be able to assign a specific dollar value to the possible outcomes.

Other times, in the case of a model, you may need to assign a value or score that represents monetary amounts. The assigned value of each outcome will be positive if you expect to earn money and negative if you expect to lose.

Determine the probability of each outcome. In a situation like the stock market, professional analysts spend their entire careers trying to determine the likelihood that any given stock will go up or down on any given day.

The probability of the outcomes usually depends on many external factors. Statisticians will work together with market analysts to assign reasonable probabilities to prediction models.

Multiply each outcome value by its respective probability. Use your list of all possible outcomes, and multiply each value times the probability of that value occurring.

Add together all the products. Find the EV for the given situation by adding together the products of value times probability, for all possible outcomes.

Interpret the results. You need to read the statistical calculation of the EV and make sense of it in real world terms, according to the problem.

Earning Method 3 of Familiarize yourself with the problem. Before thinking about all the possible outcomes and probabilities involved, make sure to understand the problem.

A 6-sided die is rolled once, and your cash winnings depend on the number rolled. Rolling any other number results in no payout.

This is a relatively simple gambling game. Because you are rolling one die, there are only six possible outcomes on any one roll.

They are 1, 2, 3, 4, 5 and 6. Assign a value to each outcome. This gambling game has asymmetric values assigned to the various rolls, according to the rules of the game.

For each possible roll of the die, assign the value to be the amount of money that you will either earn or lose. In this game, you are presumably rolling a fair, six-sided die.

Use the table of values you calculated for all six die rolls, and multiply each value times the probability of 0. Now consider a weightless rod on which are placed weights, at locations x i along the rod and having masses p i whose sum is one.

The point at which the rod balances is E[ X ]. Expected values can also be used to compute the variance , by means of the computational formula for the variance.

A very important application of the expectation value is in the field of quantum mechanics. Thus, one cannot interchange limits and expectation, without additional conditions on the random variables.

A number of convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below. There are a number of inequalities involving the expected values of functions of random variables.

Retrieved Wiley Series in Probability and Statistics. The American Mathematical Monthly. English Translation" PDF. A philosophical essay on probabilities.

Dover Publications. Fifth edition. Deighton Bell, Cambridge. The art of probability for scientists and engineers. Sampling from the Cauchy distribution and averaging gets you nowhere — one sample has the same distribution as the average of samples!

Brazilian Journal of Probability and Statistics. Edwards, A. F Pascal's arithmetical triangle: the story of a mathematical idea 2nd ed.

JHU Press. This formula makes an interesting appearance in the St. Petersburg Paradox. Share Flipboard Email. Courtney Taylor. Professor of Mathematics.

Courtney K. Taylor, Ph. Updated January 14, ThoughtCo uses cookies to provide you with a great user experience. By using ThoughtCo, you accept our.

### Expected Value Computation - How to Get Best Site Performance

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