Suppose I have a random variable X with known mean and standard deviation. After n realizations of X, what is the expected maximum of those n realizations?
When n is very large, we know the mean of those realizations will be the mean of the distribution. Is the expected maximum simply the...
Interesting, I had wondered if that was okay to do...as the variance of X would then be the sum of the variances of the two normal distributions...and this is, in fact, sqrt(2). Could you explain how these are equivalent pictures?
In general, I would like to consider a set of distributions...
Could you elaborate a bit more? I can confirm that the std deviation is indeed sqrt(2), however, I don't understand where the following formula comes from:
E(X^2 | E(x) = m) = 1 + E(m^2)
From the definition,
\sigma_x^2 = E(x^2) - E(x)^2 = E(x^2) - m^2
Presumably, I stick your formula into...
Let X be a random variable with mean \mu and standard deviation 1.
Let's add a twist.
Suppose \mu is randomly distributed about 0 with standard deviation 1.
At each iteration, we select a new \mu according to its distributuion. This mean is then used in the distribution for X. Then we pick...
Here are some thoughts. Please critque them.
When speaking informally, we write the microstate as:
(for distinct coins)
HH
HT
TH
TT
However this just shorthand as marcusl implied. In physics, we cannot specify a microstate as above.
We are treating the coins as independent. In...
You have implicitly assumed the coins are distinct. If the coins are identical, then there are only 3 microstates. This is exactly what Bose-Einstein statistics is about. From http://en.wikipedia.org/wiki/Satyendra_Nath_Bose:
"Because photons are indistinguishable from each other, one...
There is no way that this is correct. The probability of getting a single head when tossing two real coins is always 1/2 and never 1/3.
This is _interesting_. Even if I personally am unable to distinguish two coins from one another (thus, they are identical to me and to what I care about)...
Maybe this didn't come through clearly.
Why are the two coins distinct?
I guess I was thinking that distinguishablity was a function of my interest. If I am only interested in the face of the coin, then all coins are identical. Obviously this line of thinking is wrong. I promise, I'm not...
I'm looking for some insight/clarification on the definition of a microstate.
Consider two coins. Here are the possible outcomes:
T T
T H
H T
H H
Now, I have assumed something: The coins are distinct.
A better description of each microstate is:
{ (coin 1, T), (coin 2, T) }...
Wonderful! So, it doesn't appear that there is a closed form solution to this. However, the recurrence relation is quite nice, along with the Euler triangle.
Thanks!
Suppose I have 4 bosons in a one-dimensional harmonic oscillator potential and that the total energy is E_\text{tot} = 8 \hbar \omega. Recall, E_n = (n+1/2)\hbar\omega.
Question: How many quantum states exist? (assume no spin degeneracy)
After accounting for the ground state, we have 6...
Here is a much better statement of my question:
How many integer solutions exist to the following equation:
\sum_{i=1}^k n_i = N
Let me call this number p(N,k). It is the number of partitions for N such that the partitions are restricted to be of order k or less.
Example:
N = 5
k = 3...