# Thread: Normal distribution and unit

1. ## Re: Normal distribution and unit

And why do physicists can use Log-Log graph to plot Log(Dose Rate) and Log(distance)?

http://spacemath.gsfc.nasa.gov/weekly/7Page75.pdf

Distance is not unitless.

I guess this can be done only on graphs, and not on equations or physical laws, right?

2. ## Re: Normal distribution and unit

Originally Posted by eulerian
Thank you so much again.

I get it.

But, I have not learned information entropy. Could you just show out some dimensional analysis about the operation between any other units and the units of information(e.g. bits or nats)?
well... I don't want to get in over my head w/out having reviewed some of this stuff I haven't done in 30 yrs.

but basically entropy is the expected value of the log of the distribution.

for a discrete rv X, H(X) = Sum[pk ln(pk)] and the units are nats because we used the natural log. Use log2 for bits

for a continuous rv you have differential entropy given by h(X) = Integral[p(X) ln(p(x)) dx, over the support of X]

It's a measure of how random a distribution is. A drv X ~ {0.8, 0.1, 0.05, 0.05} for example has less entropy than Y ~ {0.25, 0.25, 0.25, 0.25}

It's also a measure how how expensive it is to represent the underlying random variable in terms of units of information. It takes less information to represent low entropy sources. It takes more to represent high entropy sources. This is important for compression and other types of efficient source coding.

I'll leave it to you to investigate further.

3. ## Re: Normal distribution and unit

Originally Posted by eulerian
And why do physicists can use Log-Log graph to plot Log(Dose Rate) and Log(distance)?

http://spacemath.gsfc.nasa.gov/weekly/7Page75.pdf

Distance is not unitless.

I guess this can be done only on graphs, and not on equations or physical laws, right?
if you want to get technical about it you're not representing distances or other physical things, you are representing the ratio of them to some reference point. Those ratios are unitless.

4. ## Re: Normal distribution and unit

Thank you so much. You help me a lot. I can get the rest of them.

You are highly appreciated.

5. ## Re: Normal distribution and unit

Oh, I forget one thing. What should be the unit of the sigma in this equation?

1/(units of underlying rv)^n?

Thank you.

6. ## Re: Normal distribution and unit

Originally Posted by eulerian
Oh, I forget one thing. What should be the unit of the sigma in this equation?

Thank you.
well $\displaystyle \sigma$ = sqrt[E[(X-$\displaystyle \mu$)2]].

X - $\displaystyle \mu$ has the units of whatever X is

so the square of it has those units squared. The expected value has the same units, and then the square root brings you back to the original units of X.

so $\displaystyle \sigma$ has the same units as X does.

7. ## Re: Normal distribution and unit

Originally Posted by romsek
well $\displaystyle \sigma$ = sqrt[E[(X-$\displaystyle \mu$)2]].

X - $\displaystyle \mu$ has the units of whatever X is

so the square of it has those units squared. The expected value has the same units, and then the square root brings you back to the original units of X.

so $\displaystyle \sigma$ has the same units as X does.
Yes, I knew the operation and reasons.

I mean, is 1/(units of underlying rv)^n meaningful?

Unit of the joint pdf?

8. ## Re: Normal distribution and unit

Originally Posted by eulerian
Yes, I knew the operation and reasons.

I mean, is 1/(units of underlying rv)^n meaningful?

Unit of the joint pdf?
well, remember this is a density function.

For it to become a unitless probability it has to be integrated with dx... in this case dx1 dx2 .... dxn

each of those dx's carry units so you end up with (units of X)n

thus you need the 1/(units of X)n in the density function.

9. ## Re: Normal distribution and unit

Thank you. I have no question.

10. ## Re: Normal distribution and unit

Originally Posted by romsek
if you want to get technical about it you're not representing distances or other physical things, you are representing the ratio of them to some reference point. Those ratios are unitless.
I am still a bit confused.

I know your idea, but I do not know how to express this graph in an equation.

http://spacemath.gsfc.nasa.gov/weekly/7Page75.pdf

11. ## Re: Normal distribution and unit

Also, can we do the below?

1. Y is in the unit of meter. So, we cannot log Y. But, if we transform log(Y) into log(Y/1), "1" in the unit of meter, can log(Y/1) be workable? Because the "Y/1" is unitless. And finally, we multiply antilog of log(Y/1) by 1 meter.

2. R = 20 in meters, we cannot log (20 meters), but can we log (20)(meters)?

12. ## Re: Normal distribution and unit

Also, I found some unit problems in Greeks in mathematical finance.

The Gamma should be in the unit of 1/$. http://upload.wikimedia.org/math/d/2...e055c27351.png Greeks (finance) - Wikipedia, the free encyclopedia But the unit of Gamma in the below equation seems not to be 1/$. Why?