Mean, also known as expected value, is a measure defined by mathematical expectation, which is used to describe the central position of random variables.
The expected value of binomial distribution is:
E(X) = np
Where E(X) represents the expected value of the random variable x, n represents the total number of independent experiments, and p represents the probability of success of each experiment. It can be seen from this formula that the expected value is equal to the product of the number of trials n and the probability of success p.
Consider an example. If we toss a fair coin 10 times, five of which are heads up, then the probability of success in each experiment is 0.5. In this case, the average value of the binomial distribution can be calculated as follows:
E(X) = 10 × 0.5 = 5
Therefore, when we toss a fair coin 10 times, we expect the number of heads up to be 5. This result is roughly in line with our intuition, because a fair coin is of equal probability, so the probability of heads and tails appearing is equal.
In practical application, it is very useful to calculate the mean of binomial distribution. Because the average can help us predict the average number of successful experiments, it is of great significance for making experimental plans and predicting results. At the same time, in statistical inference, we can also use the mean to evaluate the representativeness of data samples and judge whether the original data meets our expectations.