1. “Théorie Analytique des Probabilités”, 1812
“It is remarkable that a science which began with the consideration of games of chance should have become the most important object
of human knowledge.”
Pierre Simon Laplace
“Théorie Analytique des Probabilités”, 1812

2. General Discussion of Mean Values

3. P(u1), P(u2), P(u3),….., P(uM-1), P(uM)
The Binomial Distribution is only one example of a discrete probability distribution.
Now, a brief discussion of a
General Distribution.
Most of the following is valid for ANY discrete probability distribution. Let u = a variable which can take on any of M discrete values:
u1,u2,u3,…,uM-1,uM with probabilities
P(u1), P(u2), P(u3),….., P(uM-1), P(uM)

4. So, μ ≡ ū ≡ <u> ū ≡ <u> ≡ (S2/S1)
The Mean (average) value of u is:
ū ≡ <u> ≡ (S2/S1)
Here, S1 ≡ ∑iP(ui) or
S1 ≡ P(u1) + P(u2) + P(u3) +…+ P(uM-1) + P(uM)
For a properly normalized distribution, we must have:
S1 = ∑iP(ui) = 1.
We assume this from now on.
Also, S2 ≡ ∑iuiP(ui) or
S2 ≡ u1P(u1) + u2P(u2) + u3P(u3) +
…+ uM-1P(uM-1) + uMP(uM)
Note: μ ≡ Standard notation for the mean value.
So, μ ≡ ū ≡ <u>

5. <Δu> = <u - ū> = ū – ū = 0
Sometimes, ū is called the 1st moment of P(u).
If F(u) is any function of u, the mean value of F(u) is:
<F> ≡ ∑iF(ui)P(ui)
Some simple mean values that are useful for describing any probability distribution P(u):
1. The Mean Value, μ ≡ ū ≡ <u>
This is a measure of the central value of u about which the various values of ui are distributed.
Consider the quantity Δu ≡ u - ū (deviation from the mean). It’s mean is:
<Δu> = <u - ū> = ū – ū = 0
 The mean value of the deviation from the mean is always zero!

6. <(Δu)2> = <(u - <u>)2> = <u2 -2uū – (ū)2>
Now, look at (Δu)2 = (u - <u>)2 (square of the deviation from the mean). It’s mean value is:
<(Δu)2> = <(u - <u>)2> = <u2 -2uū – (ū)2>
= <u2> - 2<u><u> – (<u>)2  <u2> - (<u>)2
This is called the “Mean Square Deviation” (from the mean). It is also called several different (equivalent!) other names: The Dispersion or The Variance or the
2nd Moment of P(u) about the mean. 1
σ2 = <(Δu)2> is a measurel of the spread of the u values about ū. Also <(Δu)2> = 0 if & only if ui = ū for all i.
It can easily be shown that,
<(Δu)2> ≥ 0, or <u2> ≥ (<u>)2
Note: σ2 ≡ Standard notation for the variance.

7. <(Δu)n> ≡ <(u - <u>)n>
Could also define the nth moment of P(u) about the mean:
<(Δu)n> ≡ <(u - <u>)n>
In Physics this is rarely used beyond n = 2 & almost never beyond n = 3 or 4.
NOTE: From math: A knowledge of the probability distribution function P(u) gives complete information about the distribution of the values of u. But, a knowledge of only a few moments, like knowing just ū & <(Δu)2> implies only partial, though useful knowledge of the distribution.
A knowledge of only some moments is not
enough to uniquely determine P(u).
Math Theorem
In order to uniquely determine a distribution P(u), we need to know ALL moments of it. That is we need all moments for
n = 0,1,2,3….  .

8. Mean Values for the Random Walk Problem
In the following, as before, the results will only be summarized. See other sources for derivation details.
The Binomial Distribution is:
WN(n1) = [N!/(n1!n2!)]pn1qn2
p = the probability of a step to the right,
q = 1 – p = the probability of a step to the left.
It is easily shown that this is properly normalized:
∑(n1 = 0N) WN(n1) = 1

9. <n1> + <n2> = N(p + q) = N μ = <n1> = Np
What is the mean number of steps to the right?
<n1> ≡ ∑(n1 = 0N) n1WN(n1)
= ∑(n1 = 0N) n1[N!/(n1!(N-n1)!] pn1qN-n (1)
After derivation, the results are:
Similarly, we can also easily show that the mean number of steps to the left is:
Of course,
<n1> + <n2> = N(p + q) = N
as it should!
μ = <n1> = Np
<n2> = Nq

10. <m> = <n1> - <n2> = N( p – q)
What is the mean displacement,
<x> = <m>ℓ?
Clearly, m = n1 – n2, so
<m> = <n1> - <n2> = N( p – q)
& <x> = <m> ℓ = N( p – q)ℓ
If p = q = ½, <m> = 0 so,
<x> = <m>ℓ = = 0

11. σ2 = <(Δn1)2> = <(n1 - <n1>)2>
What is the dispersion (variance)?
σ2 = <(Δn1)2> = <(n1 - <n1>)2>
in the number of steps to the right?
That is, what is the spread in n1 values about <n1>? After some math, we find: ;
This is the dispersion or variance of the binomial distribution. The root mean square (rms) deviation from the mean is defined in general as:
σ  [<(Δn1)2>]½.
For the binomial distribution, this is
σ = [Npq]½
Note that σ  Distribution Width
σ2 = <(Δn1)2> = Npq

12. σ2 = <(Δn1)2> = Npq Summary: For the Binomial Distribution
Dispersion or variance in number of steps to the right:
Root mean square (rms) deviation from the mean:
σ  [<(Δn1)2>]½ = [Npq]½
and σ  Distribution Width
Again note that: μ = <n1> = Np. So, the relative width of the distribution, / is:
(/ ) = [Npq]½(Np) = (q½)(pN)½
If p = q, this is: (/) = 1(N)½ = (N)-½
 As N increases, the mean value increases
 N but the relative width decreases  (N)-½
σ2 = <(Δn1)2> = Npq

13. <(Δm)2> = 4<(Δn1)2>
What is the dispersion in the displacement x?
<x2> = <(Δm)2>ℓ2 = <(m - <m>)2>ℓ2
Or, what is the spread in m values about <m>?
Some math gives:
<(Δm)2> = 4<(Δn1)2>
Using <(Δn1)2> = Npq, this becomes: L
If p = q = ½, we have
<(Δm)2> = N
<(Δm)2> = 4Npq

14. Relative Width: (/ ) = (q½)(pN)½
For the 1 Dimensional Random Walk Problem
The Probability Distribution is Binomial:
WN(n1) = [N!/(n1!n2!)]pn1qn2
Mean number of steps to the right:  = <n1> = Np
Dispersion in n1: 2 = <(Δn1)2> = Npq
Relative Width:
(/ ) = (q½)(pN)½
for N increasing, the mean
value increases  N, & the
relative width decreases  (N)-½
N = 20
p = q = ½
q = 1 – p
n2 = N - n1