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Can anybody explain to me how to get the mean and the Variance for a specific function.

Thanks alot.

Thanks alot.

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- Thread starter T.Engineer
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- #1

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Can anybody explain to me how to get the mean and the Variance for a specific function.

Thanks alot.

Thanks alot.

- #2

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Can anybody explain to me how to get the mean and the Variance for a specific function.

Thanks alot.

If you know exactly the pdf (probability density function) [tex]f(x)[/tex], the formula for the mean is

[tex]\mu = E[x] = \int x f(x) dx[/tex]

and for the variance

[tex]\sigma^{2} = E[(x - E[x])^{2}] = \int (x - E[x])^{2} f(x) dx[/tex]

If you only have experimental data, you can estimate the mean and variance of the distribution :

[tex]m = 1/N\times\sum_{i = 1}^{N} x_{i}[/tex]

[tex]s^{2} = 1/(N - 1)\times\sum_{i = 1}^{N} (x_{i} - m)^{2}[/tex]

Hope it helps

- #3

EnumaElish

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For a specific function h of a random variable x with p.d.f. f(x),

Mean = E[h(x)] = ∫h(x)f(x) dx

Variance = E[(h(x) - Mean)^2] = ∫(h(x) - Mean)^2 f(x) dx

both integrated over the domain of f(x).

m = Σ_{i} h(x_{i})/N

s^2 = Σ_{i} (h(x_{i}) - m)^2/(N-1)

Mean = E[h(x)] = ∫h(x)f(x) dx

Variance = E[(h(x) - Mean)^2] = ∫(h(x) - Mean)^2 f(x) dx

both integrated over the domain of f(x).

m = Σ

s^2 = Σ

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- #4

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For a specific function h of a random variable x with p.d.f. f(x),

Mean = E[h(x)] = ∫h(x)f(x) dx

Variance = E[(h(x) - Mean)^2] = ∫(h(x) - Mean)^2 f(x) dx

both integrated over the domain of f(x).

m = Σ_{i}h(x_{i})/N

s^2 = Σ_{i}(h(x_{i}) - m)^2/(N-1)

for my function

Hn(t)= (-1)^n cos(2π fc t)* e^[(t^2)/4] *d^n/dt^n *e^[(t^2)/4]

what it will be h(x) and f(x)?

Thanks alot!

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- #5

EnumaElish

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For me to answer this, you should tell me what is random. (You need a random variable for this to work.) Are signal times (t) random? Is the time between two signals random? What is your random variable?

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- #6

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For me to answer this, you should tell me what is random. (You need a random variable for this to work.) Are signal times (t) random? Is the time between two signals random? What is your random variable?

n is also random variable.

- #7

chroot

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Please post the complete problem, exactly as it was given to you. You seem to be leaving out a lot of important information.

- Warren

- #8

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Please post the complete problem, exactly as it was given to you. You seem to be leaving out a lot of important information.

- Warren

I'd like to find the mean and variance for the following function

Hn(t)= (-1)^n cos(2π fc t)* e^[(t^2)/4] *d^n/dt^n *e^[(t^2)/4]

where n=1,2,...,N

fc=6.5MHz

- #9

EnumaElish

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You can simulate it for a given t and random n.

You can also simulate it with random t and random n.

You can collect the data and calculate the mean and the variance.

- #10

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You can simulate it for a given t and random n.

You can also simulate it with random t and random n.

You can collect the data and calculate the mean and the variance.

Thanks alot!

but I dont know how to start?

should I use the method which represented by

http://w3eos.whoi.edu/12.747/notes/lect06/l06s02.html

and if yes, how to enter my function to this simulation?

for example in the first equation , what did he mean by

yi, y

- #11

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first of all I want to find a mathematical expression for my function Hn(t).

Or , should firstly to find the data? and after that to find the mathematical expression for mean and variance of Hn(t)??

- #12

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You can also simulate it with random t and random n.

You can collect the data and calculate the mean and the variance.

And sure I prefer to simulate with random t and random n.

But how?

Thanks alot!!!

- #13

EnumaElish

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yi is the i'th individual data point (function value). (y1 = first data point, y2 = second, ...)Thanks alot!

but I dont know how to start?

should I use the method which represented by

http://w3eos.whoi.edu/12.747/notes/lect06/l06s02.html

and if yes, how to enter my function to this simulation?

for example in the first equation , what did he mean by

yi, y

"y bar" is the mean yi, calculated as the average of all the yi's:

y bar = Σ

To simulate the function, you need to answer:

1. What variables are random?

2. Are they independent?

3. What is the probability distribution function for each random variable?

Your answers may be:

1. t and n (see footnote)

2. Yes

3. This is the difficult question. What determines the time at which the signal is emitted? Is it a random process like nuclear (radioactive) decay? And what determines n?

To start simple, you can assume t

1. Let i = 0. Assume t

2. Let i = i + 1. Generate uniform random value t

3. Generate uniform random value n

4. Evaluate H[n

5. Go to step 2.

_______________________________

Footnote: Although I don't understand why n is random, I am going with your statement that n is random.

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- #14

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which it will be transmitted according to time hopping format for kth users and given by :

S(t)= [tex]\sum^{\infty}_{j=-\infty} A^k Hn(t - jTf-cj Tc - rd^kj[/tex]

where A: is the signal amplitude

Hn(t): transmitted signal

Tf: is the frame time, which is typically a hundre to a thousand times

the impulse width resulting in a signal with very low duty cycle.

Each frame is divided into N tim slots with duration Tc

cj: time-hopping sequence (0<=cj<= N) with period Tc

This provides an additional shift in order to avoid catastrophic

collisions due to multipl access interference.

d: is the sequence of the MN-ary data stream generated by the kth

source after channel coding.

r :is the additional time shift utilized by the N-ary pulse positio

modulation.

I dont know if the above information is important for what I am going to determine?

thanks alot!

- #15

EnumaElish

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It will take me some time to digest this information.

I thought Hn was only a function of t. See your earlier post https://www.physicsforums.com/showpost.php?p=1390181&postcount=4

How does the t in your last post relate to the t in your earlier post? Are they the same t? Do you mean to say Hn(#) = (-1)^n cos(2π fc #)* e^[(#^2)/4] *d^n/d#^n *e^[(#^2)/4] for some generic (general) argument # where # = t - jTf - cj Tc - r d^kj ?

If this is not it, what is it?

Assuming this is it, I advise you start simple by assuming t is uniformly distributed; you can easily change it later and replace it with a more complicated frequency distribution. I still do not understand why n is random; but if you think it is, then I am not going to argue with you. I will advise that you start simple and also assume n has a uniform frequency distribution.

Once you attach each of t and n to a frequency distribution, you can easily simulate your function to calculate the AC coefficient. You can also determine it analytically, by applying the formulas under this thread and under this other thread.

I thought Hn was only a function of t. See your earlier post https://www.physicsforums.com/showpost.php?p=1390181&postcount=4

How does the t in your last post relate to the t in your earlier post? Are they the same t? Do you mean to say Hn(#) = (-1)^n cos(2π fc #)* e^[(#^2)/4] *d^n/d#^n *e^[(#^2)/4] for some generic (general) argument # where # = t - jTf - cj Tc - r d^kj ?

If this is not it, what is it?

Assuming this is it, I advise you start simple by assuming t is uniformly distributed; you can easily change it later and replace it with a more complicated frequency distribution. I still do not understand why n is random; but if you think it is, then I am not going to argue with you. I will advise that you start simple and also assume n has a uniform frequency distribution.

Once you attach each of t and n to a frequency distribution, you can easily simulate your function to calculate the AC coefficient. You can also determine it analytically, by applying the formulas under this thread and under this other thread.

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- #16

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Are they the same t? Do you mean to say Hn(#) = (-1)^n cos(2π fc #)* e^[(#^2)/4] *d^n/d#^n *e^[(#^2)/4] for some generic (general) argument # where # = t - jTf - cj Tc - r d^kj ?

If this is not it, what is it?

yes, exactly! that's right!

- #17

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I will advise that you start simple and also assume n has a uniform frequency distribution.

You mean firstly I will work for n=1, for example.

is not that right?

- #18

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you mean to say Hn(#) = (-1)^n cos(2π fc #)* e^[(#^2)/4] *d^n/d#^n *e^[(#^2)/4] for some generic (general) argument # where # = t - jTf - cj Tc - r d^kj

Once you attach each of t and n to a frequency distribution, you can easily simulate your function to calculate the AC coefficient. You can also determine it analytically, by applying the formulas under this thread and under this other thread.

Now, can you tell me how to start and from where?

Really I get confused.

Thanks alot!

- #19

EnumaElish

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See my last post under this new thread: https://www.physicsforums.com/showthread.php?t=179061

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