6.6:
Expected Value
Consider a probability distribution obtained by rolling a die a hundred times. The mean is calculated using its formula.
As n increases, the mean value fluctuates, but as seen in this graph of mean versus the number of trials, the mean gradually approaches a constant value with increasing trials.
The expected value of a random variable is the mean value as the sample size grows to infinity. In simple words, it is the long-run average of the results.
So, its formula is similar to that of the mean.
The concept of expected value is useful in decision theory. If one bets ten dollars on number 8 in roulette, there are 37 of 38 chances to lose and one of 38 chances to win.
If the winning money on the table is 360 dollars, the net gain on this small chance event would be 350 dollars.
The product of the random variable, with its probability, is summed to obtain the expected value.
This number tells us that one can expect to lose 53 cents for every ten-dollar bet.
6.6:
Expected Value
The expected value is known as the "long-term" average or mean. This means that over the long term of experimenting over and over, you would expect this average. The expected average is represented by the symbol μ. It is calculated as follows:
In the equation, x is an event, and P(x) is the probability of the event occurring.
The expected value has practical applications in decision theory.
This text is adapted from Openstax, Introductory Statistics, Section 4.2 Mean or Expected Value and Standard Deviation.