Difference between mean and expected value

・In Japanese
＜Premise knowledge＞
・mean value

■Expected Value

The expected value is the mean value that is likely to be obtained if the trial is performed. For example, the mean value of the amount of money obtained in the lottery, the mean value of the dice roll, etc. can be expressed as the expected value. The formula is below.



For example, when the probability of a dice roll is as follows, the expected value of the dice roll is calculated.



The calculation formula is as follows.

■Difference between expected value and mean value

the expected value is a kind of mean value. The difference is that the expected value is an anticipated mean value for future trials, whereas it is the mean value for the results of previous trials. Also, if the expected value p (probability of occurrence) is uniform, the expected value formula is the same as the mean value.



To explain the example of rolling the dice, the mean value of the rolled rolls that can be calculated before rolling the dice is the expected value, and the mean value when calculating from the rolled rolls after actually rolling the dice. The more times you roll the dice, the more the mean value will converge to the expected value.

＜Example＞
When the expected value was (1) above, the number of rolls rolled three times was 5,4 and 3. What is the mean value of the dice rolls?

■Expected value trap

For example, if you have a 90% chance of winning 10 million yen, a 10% chance of winning 100 million yen, or a one-time lottery, which lottery would you choose?

Some people may think, "Choose a lottery with a high expected value of 100 million yen", but the expected value is effective when the number of trials is large, so you should not think about the expected value in one trial. In this case, there is no answer that it is theoretically correct to choose which one, but it seems that there are many people who have a desire to avoid loss and choose a lottery of 10 million yen, which is likely to be received even if the amount is small.

On the flip side, it may be better to choose the one with less psychological stress.

■Expected value symbol

The expected value can be expressed by E (Expectation) as shown above. In some cases, a polynomial is put in E and the expected value (or mean value) of the polynomial is calculated as shown below. For example, when finding the variance.



The mean value can be expressed by the X-bar below. Often synonymous with E, X-bar is used.

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