What do moments mean on path

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Due in large part to merchandising concerns, many celebrities have attempted and often succeeded in trademarking far more mundane words—for example, Taylor Swift holds trademarks for "Nice to Meet You.

Where You Been? You can absolutely pave the way for one, by staying open to what you can learn from life's curveballs. Many of Oprah's aha moments came after what, at first glance, looked like a major setback—such as her move from news anchor to talk show co-host as a young reporter in Baltimore.

After a perspective shift, her realization came. Reading one of Oprah's favorite pastimes is another great way to unearth an aha moment, because sitting with a great author's thoughts can so often inspire your own.

Oprah agrees. It's a resonating with what is somehow buried, or suppressed. 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. Caution: 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].

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 , :.