Difference Between Mean and Expectation
Average or mean is a very common concept in mathematics and statistics. There is the arithmetic mean is more popular and taught in the younger classes, but there is also the expected value of a random variable known as the population mean and is a part of statistical studies in higher classes. Arithmetic and expectation, the two types of means are similar in nature although they have also some differences. Let us know these differences by emphasizing on basic features of both.
The concept of expectation came into existence because of gambling and this often became a problem when a game ended with no logical end as the players could not do the distribution of stakes agreeably.
While the mean is the simple average of all values, the expectation of the expected value is the average of a random variable weighted with probability. Expectation can be understood by the concept of tossing a coin 10 times. When you flip the coin for 10 times, you expect five heads and five tails. This is known as the value of expectation because the chances of getting a head or a tail in every toss are 0.5. If you say the heads, the chances of receiving a head is 0.5, the expected value for 10 tosses is 0.5 1x 0 = 5. Thus if p is the chance of an event occurring and there are n number of events, the mean will be equal to n x p. In cases where random variable X is real valued, the expectation value and mean will be same. While means ignores probability, expectation is probability-weighted. The fact that the expectation is described as the weighted mean or average of all possible values that a random variable can take, the expectation is quite different from mean, which is simply the sum of all values divided by the number of values.