How to Calculate Mean & Variance of a Function

[T.Engineer]

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

 

[Barmecides]

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) $$f(x)$$, the formula for the mean is
$$\mu = E[x] = \int x f(x) dx$$
and for the variance
$$\sigma^{2} = E[(x - E[x])^{2}] = \int (x - E[x])^{2} f(x) dx$$

If you only have experimental data, you can estimate the mean and variance of the distribution :
$$m = 1/N\times\sum_{i = 1}^{N} x_{i}$$
$$s^{2} = 1/(N - 1)\times\sum_{i = 1}^{N} (x_{i} - m)^{2}$$

Hope it helps

 

[EnumaElish]

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)

 

[T.Engineer]

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!

 

[EnumaElish]

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?

 

[T.Engineer]

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.

 

[chroot]

T.Engineer,

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

- Warren

 

[T.Engineer]

T.Engineer,

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

 

[EnumaElish]

You can simulate this for a given n (random t).

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.

 

[T.Engineer]

You can simulate this for a given n (random t).

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 don't 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

 

[T.Engineer]

Also I'd like to find an expression for the mean and variance in general not by just data.
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)??

 

[T.Engineer]

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!

 

[EnumaElish]

Thanks alot!
but I don't 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

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

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

y bar = Σ_i yi / N for i = 1, ..., N.

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_i is distributed uniformly between t_i-1 and T_i, where T_i is an upper bound. Also assume n is uniformly distributed between 0 and M (a large number).

1. Let i = 0. Assume t₀ = 0. Assume n₀ = 0.
2. Let i = i + 1. Generate uniform random value t_i between t_i-1 and T_i (say, T_i = t_i-1 + 1)
3. Generate uniform random value n_i between 0 and M (say, M = 10)
4. Evaluate H[n_i](t_i).
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.

 

[T.Engineer]

The transmitted signal represented by the function Hn(t)
which it will be transmitted according to time hopping format for kth users and given by :

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

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 don't know if the above information is important for what I am going to determine?
thanks alot!

 

[EnumaElish]

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.

 

[T.Engineer]

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!

 

[T.Engineer]

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?

 

[T.Engineer]

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!

 

[EnumaElish]

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