The concept and nature
definition
Expectation is that probability theory a very important concept. If X is a discrete random variable distribution as p (x), then X is referred to as the desired E [X], is defined as:
If X is a continuous random variable with probability density function f (x), then the expectation E X [X] is defined as:
Expression of a desired language, X is a weighted average of all of the possible X values, the weight of each weight value is the probability that the value X is taken. For chestnut:
If X is distributed as
p(0) = 1/2 = p(1)
Then
E[X] = 0 * 1/2 + 1 * 1/2 = 1/2
This is the average value of X of two possible values 0 and 1 in the usual sense. On the other hand, if the
p(0) = 1/3, p(1) = 2/3,
Then
E[X] = 0 * 1/3 + 1 * 2/3 = 2/3
This is a weighted average of two possible values 0 and 1, because p (1) = 2p (0), at this time a weight of 1 is 2 times the weight zero.
Expect common distribution
Mathematical expectation E [X] by the probability of a random variable X is completely determined by the distribution, subject to a distribution if X, also known as E [X] is the distribution of mathematical expectation.
A desired common-dimensional random variables as follows:
0--1 Distribution:
E[X] = p
Binomial distribution, X ~ B (n, p):
E [X] = np
Poisson distribution, X ~ P (λ):
E [X] = k
Uniform distribution, X ~ U (a, b):
E[X] = (a + b) / 2
Exponential distribution, X ~ E (λ):
E [X] = 1 / l
The normal distribution, X ~ N (μ, σ ^ 2):
E [X] = μ
Following derivation desired uniform distribution and the Poisson distribution:
Poisson distribution:
Evenly distributed:
For chestnuts