The expectation (mean or the first moment) of a discrete random variable X is and the second column lists the probabilities f(x) associated with these values. The moments of a distribution generalize its mean and variance. Its second moment about the origin, denoted mjpg Similar relationships can be found between the higher moments by writing out the terms of the. center of the distribution of X. The variance of X is a measure of the spread of the distribution about the mean and is Thus, the variance is the second moment of X about .. connection can help unify and illuminate some of the ideas.
Its second moment about the origin, denotedis defined as the expected value of the random variable X2, or E[X2]. In general, the rth moment of X about the origin, denotedis defined as. The third moment about the mean,is used to construct a measure of skewness, which describes whether the probability mass is more to the left or the right of the mean, compared to a normal distribution.
The two definitions of a moment are related. Moment Generating Functions Except under some pathological conditions, a distribution can be thought to be uniquely represented by its moments. That is, if two distributions have the same moments, they will be identical except under some rather unusual circumstances.
Moments: Mean and Variance
Such an expression should have terms corresponding to for all values of r. We can get a hint regarding a suitable representation from the expansion of ex: We see that there is one term for each power of x. This suggests the definition of the moment generating function of a random variable X as the expected value of etX, where t is an auxiliary variable: To see how this represents the moments of a distribution, we expand M t as Click to view larger image Thus, the MGF represents all the moments of the random variable X in a single compact expression.
Note that the MGF of a distribution is undefined if one or more of its moments are infinite.
Moments: Mean and Variance | STAT
We can extract all the moments of the distribution from the MGF as follows: If we differentiate M t once, the only term that is not multiplied by t or a power of t is. Generalizing, it is easy to show that to get the rth moment of a random variable X about the origin, we need to differentiate only its MGF r times with respect to t and then set t to 0.
The exponential is merely a convenient representation that has the property that operations on the series as a whole result in corresponding operations being carried out in the compact form. For example, it can be shown that the series resulting from the product of Click to view larger image This simplifies the computation of operations on the series. However, it is sometimes necessary to revert to the series representation for certain operations.Distribution moments
One path to this result involves the distribution's characteristic function, which can be expressed by Taylor series expansion of the exponential thus yielding an infinite sum of moments: In turn, the jth moment may be recovered from the characteristic function by taking its jth derivative with respect to t followed by the limit as t goes to zero, and then multiplying by -i j.
In the equations of this section only, i is the square root of minus one. The figure above lists the first few moments, as well as the characteristic function, for the discrete Poisson distribution by way of example.
Thus the moments give us the characteristic function of the probability distribution. Can we use this characteristic function to determine the probabilities themselves? One possibility is to note that, using the "classical physics" convention for defining Fourier transforms, the characteristic function is the complex conjugate of P[t], the continuous Fourier transform of p[x].
Second moment method - Wikipedia
Inverse transforming, and denoting complex conjugation or Fourier phase reversal with an overbar, we might therefore say that: Thus the characteristic function is little more than the probability distribution's Fourier transform.
By way of example, consider the characteristic function for the univariate normal distribution given in the normal distribution figure above.
The Fourier transform of its complex conjugate is none other than the univariate normal distribution itself, i. If only discrete values of x are allowed, per the summation notation above, the result should therefore be a sum of Dirac delta functions that are only non-zero at those values. Can you show this to be true for the Poisson distribution? By combining the first two expressions in this section, can we come up with a more direct expression for p[x] in terms of the moments?
Does the result have convergence problems?
Does it also bear a relationship to the expression below? An interesting but perhaps more abstract expansion of probabilities, in terms of the central moments, is discussed in the American Journal of Physics article by Daniel Gillespie AJP 49, Although the above expression appears to be quite explicit, turning it into an explicit formula for p as a function of x can be non-trivial since derivatives of the delta function are normally defined by what happens when one integrates them over x.
Hence some examples of this, in action, could help out a lot. Gillespie takes this discussion further by noting that one can approximate derivatives of the delta function via the Gaussian distribution as a limit.
This takes him to the odd assertion here: Moment expansions of KL divergence KL-divergence involves the comparing of two probability distributions, one p that we'll call actual and another po that we'll here call ambient or reference.
Here the subscript p on the angle brackets denotes an average over the actual pi rather than the ambient poi probabilities. Note also that use of the natural log rather than log to the base 2 yields information units of nats rather than bits. The figure below depicts a single bivariate two-layer distribution.