# Redshift Academy   Wolfram Alpha: Search by keyword: Astronomy

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- Units, Constants and Useful Formulas

- ```Expected Value
--------------

The expected value is also known as the average, mean
or first moment.  It is defined as:

N
E = ΣxiP(xi)
i
Where x is a random variable and P(x) is the probability
of occurence.

Example:  Dice.

1 + 2 + 3 + 4 + 5 + 6 = 21  Average = Σx/Nn= 21/6 = 3.5

1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5

Variance
--------
_
s2 = E[(x - x)2]
_
≡ Σ(x - x)2/N
_    _
= E[x2 - 2xx + x2]
_       _
= E(x2) - 2xE(x) + x2E(1)
_
= E(x2) - 2E(x)E(x) + E(x)E(x)E(1)  since x = E(x)

= E(x2) - E(x)E(x)

= Σx2/N - (Σx/N)2

= NΣx2 - (Σx)2
------------
N2

This is the computational form for the variance.

Skewness
--------
_
Σ(xi - x)3/N           _
g1 =  ------------ = E[(xi - x)3/s3]
s3

Kurtosis
--------
_
Σ(xi - x)4/N           _
g2 =  ------------ = E[(xi - x)4/s4]
s4
``` 