according the definition of the expected value, it also refers to mean. But in scipy.stats.binom, they get different values. like this,
import scipy.stats as st
st.binom.mean(10, 0.3) ----> 3.0
st.binom.expect(args=(10, 0.3)) ---->3.0000000000000013
so that makes me confusing!! why?
In the example the difference is in floating point computation as pointed out. In general there might also be a truncation in expect depending on the integration tolerance.
The mean and some other moments have for many distribution an analytical solution in which case we usually get a precise estimate.
expect is a general function that computes the expectation for arbitrary (*) functions through summation in the discrete case and numerical integration in the continuous case. This accumulates floating point noise but also depends on the convergence criteria for the numerical integration and will, in general, be less precise than an analytically computed moment.
(*) There might be numerical problems in the integration for some "not nice" functions, which can happen for example with default settings in scipy.integrate.quad
This could be simply a result of numerical imprecision when calculating the average. Mathematically, they should be identical, but there are different ways of calculating the mean which have different properties when implemented using finite-precision arithmetic. For example, adding up the numbers and dividing by the total is not particularly reliable, especially when the numbers fluctuate by a small amount around the true (theoretical) mean, or have opposite signs. Recursive estimates may have much better properties.
