Moment about the mean
The kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity E[(X-E[X])k], where E is the expectation operator. Some random variables have no mean, in which case the moment about the mean isn't defined. The kth moment about the mean is often denoted μk. For a continuous univariate distribution P(x) the moment about the mean μ is<math> \mu_k
\left\langle ( x - \langle x \rangle )^k \right\rangle
\int_{-\infty}^{+\infty} (x - \mu)^k P(x)\,dx </math>Sometimes it is convenient to convert moments about the origin to moments about the mean. The general equation for converting the nth order moment about the origin to the moment about the mean is
<math> \mu_n = \sum_{j=0}^n {n \choose j} (-1) ^{n-j} \mu'_j m^{n-j}, </math>
where m is the mean of the distribution, and the moment about the origin is given by
<math> \mu'_j = \int_{-\infty}^{+\infty} x^n P(x)\,dx. </math>
The first moment about the mean is zero. The second moment about the mean is called the variance, and is usually denoted σ2, where σ represents the standard deviation. The third and fourth moments about the mean are used to define the standardized moments which are in turn used to define skewness and kurtosis, respectively.
And I,--I am alive, and a part of thee!
AN IDYL OF THE MAY.
Lapsing soon into June;
Of the green and go/golden.html">golden May,
To the coming feet of June,
Gleamed the ash buds through the darkness
And the waters of the stream
Like a silver.html">silver-footed dream--
Flashing, falling,
Slipping down.html">down.html">down,
Follow--follow!
Lit with patches of the sky,
Hand in hand we went together,
Of the May;
Flashing by, called--follow--follow!
Speaking low--
To the brooklet's crystal call,
Slow, and pausing here and there
For the lovely maiden-hair.html">hair;
Calling faintly down the burn.html">burn.
Still the streamlet slid away,
To a mossy nook and brown,
Where the nodding wind-flower grows,
Brooding o'er the brooklet's brink
Sweeter than the musky scent
Perfumes into perfect blent.
In the golden, golden weather--
Stole away,
Wandering down the burn together.
Ah Valentine--my Valentine!
Grave and low, and sweet and slow,
Brooding notes, half sung half said,--
Hearts make Edens of their own;
Wilt thou.html">thou.html">thou live for me alone,
Thou and I and love together?"
"Yea," I said, "for thee alone,"--
Too much their own happiness,
Like this ferny hollow--
Wilt thou bravely follow?"
Called, "Oh hear! Oh follow!"
"Where thou goest, I will go;
Through the dreariest desert land,
Noiseless as the silver dew;
In the silence spoke to each;
Shot with shafts of sunset through,
"True, till death"
Through a mist of smiles and tears,
Oh! how long ago it seems,
Silver threads are in my hair
Back along the hills of hope
.
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