If you’ve heard of probability theory, you should have come across the expected value of random variable. It’s the long-run average value of repetitions of a certain experiment. The expected value or EV for short, is also known as the mathematical expectation, mean value, average and first moment.
In more practical terms, the expected value of a discrete random variable can be defined as the probability-weighted average of all the possible values. The expected value never exists for random variables with distributions and large tails, including the Cauchy distribution.
The expected value is a number one aspect of how we characterize a probability distribution. The EV plays crucial roles in many contexts. For instance, in regression analysis, one may require a formula in terms of observed data, which could provide a decent estimate of the required parameter, giving the effect of the explanatory variable on the dependent variable. Well, the formula will provide quite different estimates with the help of different samples of data.
A typical example of utilizing the expected value for reaching rational decisions is the Gordon-Loeb Model of information security investment. In accordance with the model, the amount a company shells out in order to protect its information should be a relatively small fraction of the expected loss.
In this particular review, we are going to highlight the most common method of calculating the expected value - calculating a basic expected value problem.
The expected value is often used in statistics. That’s a good tool, when it comes to deciding how harmful or beneficial this particular action will be. In order to calculate the expected value, one requires gaining a full understanding of every outcome in the given situation and the probability of every outcome taking place. The steps illustrated here below will guide you through several typical problems to teach you the overall concept of the expected value.
First, you require familiarizing yourself with the problem. So, before you start thinking about all the outcomes as well as probabilities involved, you should make sure you properly understand the problem. For example, let’s assume a die-rolling game costs $10 per play. Then, a 6-sided dire is rolled once. Your cash winnings actually depend on the number rolled. If you roll a six, you’ll win $30. In case of rolling a 5, you’ll get $20, while you’ll win nothing in other number results.
Secondly, you should list all the possible results. In this case, you’ll be dealing with six probable results:
Keep in mind that every outcome appears to be $10 lower then described above. It’s because you require shelling out $10 in order to play the game, regardless what result you’re rolling.
Thirdly, you require determining the overall probability of every outcome. Frankly speaking, the probability of every outcome of the six available is absolutely the same. When rolling a six-sided die, your chance of a certain number being rolled is one in six. In order to make it easier to make calculations, you’d better convert this fraction into a decimal by simply plugging this stuff into a calculator. You should write the probability next to every outcome, especially if you’re working on the problem with a different probability for every outcome.
Fourthly, you should write down the value of every result. Simply multiply the $ amount of one outcome by the probability of that result in order to figure out how many evergreen bucks this particular outcome contributes to the expected value. For example, your result of rolling of one is $10 and the probability of rolling one is 0.167. In this case, the expected value of rolling one would be (-10) * (0.167).
By the way, it makes no sense to calculate all of this if you own a calculator, simultaneously supporting multiple operations. You’ll obtain a more accurate result in case of plugging in the whole equation.
Then, you require adding the value of every result together in order to obtain the expected value of the event. Going further with the example illustrated above, the EV of the dice game will be(-10 *.167) + (-10 *.167) + (-10 *.167) + (-10 *.167) + (10 *.167) + (20 *.167), or simply - $1.67. So, when playing the dice game, you require expecting to lose $1.67 per game played.
It’s up to you to properly understand all the implications of the EV calculation. In the previously mentioned example, it was determined that the expected wins of the game appeared to be - $1.67 for every roll.
That’s an absolutely unreal result for one game, so you can only win $10, lose $10 or win $20. By the way, the EV would be useful exactly as a long-term figure. If you keep playing this dice game, on average you’ll lose approximately $1.67.
It’s time to better understand the very essence of the EV. Well, the EV doesn’t necessarily need to be the desired result of yours. Sometimes, an EV will be an absolutely unreal result. For example, in this case, the EV might be + $5 for a game and the only prize of $10. So, what’s the point in this EV? OK, the EV calculates how much value you require putting on this particular event.
You should understand the very concept of independent events. Ok, in everyday life a great number of people think that they’re enjoying a lucky day if a couple of good things happen to them. People tend to expect many good things to arise. Alternatively, people might think that they’ve had enough bad luck today, and they won’t have anything bad for a while. From the mathematical point of view, events don’t always work this way.
A popular belief that you’ve got an unlucky or lucky run of coin flips is known as the gambler’s fallacy. Ti’s named after human tendency to make risky and quit foolish decisions if they believe they’re on a lucky streak.
Finally, you require understanding the law of large numbers. Most probably, you consider the EV to be absolutely useless, because it rarely point out to the result you end up with. If you determine the EV of a roulette game is - $1%, then if you play three games, most likely you’ll finish with +$60 or -$10. In fact, the law of large numbers really helps to explain why the EV is a useful thing. The secret is very simple – the more games you play, the higher the probability to get closer to the expected results.
The idea of the EV arose in the middle of the 17th century. It actually evolved from the problem of points. The problem was debated for a long time, even for centuries and a great number of solutions were suggested. In 1654 a French amateur mathematician and writer Chevalier de Méré told that the given problem could be tackled and it clearly demonstrated numerous downsides of that time’s math when it came to its practical application.
Being a mathematician, Pascal made up his mind tackle that problem. So, he started discussing the problem in a series of letter to Pierre de Fermat. Soon enough they independently worked out an effective solution. Both researchers tackled the problem in different way in terms of computation. However, their results were absolutely the same. It’s because their calculations were built around on the same principle. The essence of this principle is that the overall value of a future revenue needs to be directly proportional to the probability of getting it. Unfortunately, they didn’t publish their works.
In 1657, Christian Huygens, a Dutch mathematician published a treatise “De ratiociniis in ludo aleæ” on this popular theory. In this work he considered the problem of points and offered a solution built around the same principles as the findings of Fermat and Pascal.
By the way, Huygens also updated the concept of expectations by simply adding rules on how to figure out expectations in more difficult situations compared to the original problem.
In 1814, Pierre-Simon Laplace updated the theory of the EV.
In 1901 Whitworth was the first to use the letter «E» to denote the expected value. The symbol is still popular in this discipline.
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