Random Systems

1 Randomness

Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination.

Fig.1.Randomness in three dimensions

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(The figure and the code is from the Web)

2 Randomness for uniform distrubution

Now we get some random numbers in a two dimensional uniform distribution.

Fig.2.Randomness in two dimensions

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Now let's see randomness in two dimensions again. We draw a picture that shows that if we comtain 200 random numbers for four times, the results don't relate to each other.

Fig.3.Randomness in two dimensions(2)

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From above, we can find that as for random phenomenon, if the initial conditions are the same, there are many different results and there are little relation between them.

Fig.4.randomness

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From above, we know that if we catch a random number once after another, there are little relation between them.

Now, we use randomness to calculate a integral.
First, we use randomness to produce random points in

.

Fig.5.integral

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and then we cound the points that below the line

The results all approaches to 0.5. So the integral is 0.5.

3 Randomness for normal distrubution

The normal distrubution is

\ =\frac{1}{\sqrt{2\pi}\sigma}e ^{{-\frac{(x-\mu)}{2}}{2\sigma^{2}}})
When
the normal distrubution is called the standard normal distrybution which is showed by
\ = \frac{1}{\sqrt{2\pi }}e ^{-\frac{x{2}}{2}})

Fig.6.normal distrubution

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This is the histogram that shows the distrubution of the numbers that got from the normal distrubution which satisfies
. (The figure and the code are from the web.)

As we know,if we measure a physical quantity, there must exist error and the distrubution of the error is normal distrubution. Now we suppose that we measure the length of a box, and the results satisfies the normal distrubution that 
The results is

Fig.7.numbers from a normal distrubution

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From above, we can see that with the more n, the obviously that the distrubution of the measuremental results is normal distrubution.

4 The normal distrubution for two dimensions

The expression of the two-dimensional normal distrubution is \ =(2 \pi \sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}){-1}\ exp{\frac{1}{2 (1- \rho^{2})}[ \frac{(x-\mu_{1})^{{2}}{\sigma_{1}}{2}}+ \frac{2\rho (x-\mu_{1})(y- \mu_{2})}{ \sigma_{1} \sigma{2}}+ \frac{(y-\mu_{2})^{{2}}{\sigma_{2}}{2}}]})

If

the distrubution is written as
\ = \frac{1}{2\pi}\ e^{-(x{2} +y^{2})})
Now let's study this distrubution.

Fig.8.two dimensions for n=100

Fig.9.two dimensions for n=500

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These points are gotten from the two-dimensional normal distrubution.

5 Conclusion

I discussed the random numbers for mormal distrubution and the normal distrubution for one dimension and two dimensions. And I use them to calculate an integral and study the distrubution of measurement errors.

[1] Computational Physics(Second Edition)
[2] matplotlib
[3] LaTeX