Commit a4f26df8 authored by Maximilian Vitz's avatar Maximilian Vitz
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### 1 Exspectation Value and Variance of a Sample
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1)
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See lecture slide 46.
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2)
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$\left< V(x) \right> = \left< \frac{1}{N} \sum_{i} (x_{i}-\mu_{x})^{2} \right> = \frac{1}{N} \left< \sum_{i} x_{i}^{2} - 2 x_{i} \mu_{x} + \mu_{x}^{2} \right> = \frac{1}{N} \left( \sum_{i} \left< x_{i}^{2} \right> - 2 \left< x_{i} \right> \mu_{x} \right) + \mu_{x}^{2} = \left< x^{2} \right> - 2 \mu_{x}^{2} + \mu_{x}^{2} = \left< x^{2} \right> - \left< x \right>^{2} = \sigma_{x}^{2}$
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By fixing a parameter for the measured mean like it was done in 1) one has to apply a correction factor. If the true mean is known this is not neccessary any more.
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### 2 Poisson Distribution
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$P_{k} (T) = \frac{1}{k!} e^{- \lambda T} (\lambda T)^{k}$
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(lhs) $\dot{P}_{k}(T) = \frac{1}{k!} e^{- \lambda T} (- \lambda (\lambda T)^{k} + k \lambda^{k} T^{k-1} )$
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(rhs) $\lambda [P_{k-1}(T) - P_{k}(T)] = \lambda \left[ \frac{e^{-\lambda T}}{(k-1)!} (\lambda T)^{k-1} - \frac{e^{-\lambda T}}{k!} (\lambda T)^{k} \right] = \frac{1}{k!} e^{- \lambda T} (- \lambda (\lambda T)^{k} + k \lambda^{k} T^{k-1} )$
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$\Rightarrow \dot{P}_{k}(T) = \lambda [P_{k-1}(T) - P_{k}(T)]$
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Differential Equation was derived by comparing small time intervall with a Taylor expansion, see lecture slide 62.
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### 3 Bionomial- and Poisson Distribution
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1)
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Bionomial Distribtion for probobility for success with N trials!
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$B(k; N, p) = \left(\begin{array}{c}
N \\ k
\end{array}\right) \cdot p^{k} \cdot (1-p)^{N-k}$
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N = 60
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p = 0.01
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k = 0
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$\Rightarrow B(0; 60, 0.01) = \left(\begin{array}{c}
60 \\ 0
\end{array}\right) \cdot 0.01^{0} \cdot (1-0.01)^{60-0} = 1 \cdot 1 \cdot (0.99)^{60} = 54.7 \%$
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2)
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Poisson Distribution for event which occur with an average event rate $\lambda$.
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$P(k, \lambda) = \frac{\lambda^{k}}{k!} e^{- \lambda}$
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$\lambda$ = $0.01 \cdot 60 = 0.6$
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k = 0
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$P(0,0.6) = \frac{0.6^{0}}{0!} e^{- 0.6} = 1 \cdot e^{-0.6} = 54.9 \%$
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### 4 Poisson Distribution, Search for free Quarks
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1)
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$\sum_{i=1}^{110} P(i;229) = \sum_{i=1}^{110} \frac{229^{i}}{i!} e^{-229}= 1.6 \cdot 10^{-18} = p_{se}$
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Very low propability, very unlikely to happen!
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2)
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N = 55000
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$p_{me} = N \cdot p_{se} = 8.9 \cdot 10^{-14}$
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3)
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$\sum_{i=1}^{28} P(i;57) = \sum_{i=1}^{28} \frac{57^{i}}{i!} e^{-57}= 6.7 \cdot 10^{-6}$
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Much more likely to happend!
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4)
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Combine both Poisson Distributions to achieve the new probability:
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$\mu = 4$
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$\mu \lambda = 229$
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$\sum_{i=1}^{110} \sum_{N=0}^{\infty} P(i;N\mu) \cdot P(N;\lambda) = \sum_{i=1}^{110} \sum_{N=0}^{\infty} \frac{N \mu^{i}}{i!} e^{-N \mu} \cdot \frac{\lambda^{N}}{N!} e^{-\lambda} = 4.2 \cdot 10^{-5}$
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Even more likely to happen.
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### 5 Why are soccer results more random than handball results?
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1)
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$\lambda_{1} = 1$
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$\lambda_{2} = 2$
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$\sum_{i=1}^{\infty} \sum_{l=0}^{k-1} P(k;\lambda_{1}) \cdot P(l;\lambda_{2})= \sum_{i=1}^{\infty} \sum_{l=0}^{k-1} \frac{\lambda_{2}^{k}}{k!} e^{-\lambda_{1}} \cdot \frac{\lambda_{2}^{l}}{l!} e^{-\lambda_{2}} = 18.3 \%$
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2)
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$\lambda_{1} = 10$
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$\lambda_{2} = 20$
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$\sum_{i=1}^{\infty} \sum_{l=0}^{k-1} P(k;\lambda_{1}) \cdot P(l;\lambda_{2})= 2.6 \%$
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Summing up over all possible combinations, i=1 as at least one goal is needed to win a game.
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### 5 Python Script (thanks to Joep Geuskens)
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``` python
from scipy.stats import poisson
import numpy as np
```
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``` python
def func(l=1):
arr = np.array([poisson.cdf(k-1, 2*l)*poisson.pmf(k,l) for k in range(1,l*10)])
p = np.sum(arr)
diff = arr[-1]/p # relative size of the last term compared to the total probability
print(f"P(B>A)={p:.4f}")
print(f"Diff={diff:.2g}") # should be sufficiently small
```
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``` python
print("1)")
func(1)
print("\n2)")
func(10)
```
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1)
P(B>A)=0.1826
Diff=5.6e-06
2)
P(B>A)=0.0258
Diff=1.9e-60
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### 6 Multidimensional Gaussian
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$B =
\frac{1}{\sigma_{1}^{2}\sigma_{2}^{2}- \rho^{2}\sigma_{1}^{2}\sigma_{2}^{2}}
\left(\begin{array}{cc}
\sigma_{2}^{2} & -c\\
-c & \sigma_{1}^{2}
\end{array}\right) =
\frac{1}{\sigma_{1}^{2}\sigma_{2}^{2} \cdot (1- \rho^{2})}
\left(\begin{array}{cc}
\sigma_{2}^{2} & -c\\
-c & \sigma_{1}^{2}
\end{array}\right)$
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$c = \rho x_{1} x_{2}$
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$det B = \frac{1}{\sigma_{1}^{2}\sigma_{2}^{2} \cdot (1- \rho^{2})}$
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$k =\sqrt{\frac{det B}{(2\pi)^{n}}} = \frac{1}{2 \pi \cdot \sigma_{1}\sigma_{2} \cdot \sqrt{(1- \rho^{2})}}$
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$$
\left(\begin{array}{c}
x_{1} \\ x_{2}
\end{array}\right)
\left(\begin{array}{cc}
\sigma_{2}^{2} & -c\\
-c & \sigma_{1}^{2}
\end{array}\right)
\left(\begin{array}{cc}
x_{1} & x_{2}\\
\end{array}\right) = x_{1}^{2}\sigma_{2}^{2} - 2 x_{1} x_{2} c + x_{2}^{2}\sigma_{1}^{2} =
\sigma_{1}^{2} \sigma_{2}^{2} \left( \frac{x_{1}^2}{\sigma_{1}^{2}} + \frac{x_{2}^2}{\sigma_{2}^{2}} - 2 \rho \frac{x_{1}x_{2}}{\sigma_{1}\sigma_{2}} \right)
$$
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$\Phi (x_{1}, x_{2}) = \frac{1}{2 \pi \cdot \sigma_{1}\sigma_{2} \cdot \sqrt{(1- \rho^{2})}} \exp \left( -\frac{1}{2 \cdot (1- \rho^{2})} \left( \frac{x_{1}^2}{\sigma_{1}^{2}} + \frac{x_{2}^2}{\sigma_{2}^{2}} - 2 \rho \frac{x_{1}x_{2}}{\sigma_{1}\sigma_{2}} \right) \right)$
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Substitue $x_{i}$ with $x_{i}- a_{i}$ to achieve the exspected form.
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``` python
```
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### Combination of Measurements I
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1)
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$x_{1} = 12 \pm 3 \text{ (sta.)} \pm 2 \text{ (sys.)}$
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$x_{2} = 8 \pm 4 \text{ (sta.)} \pm 2 \text{ (sys.)}$
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For derivation please watch lecture slide 127.
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$x_{m} = \frac{x_{1}/\sigma_{1}^{2} + x_{2}/\sigma_{2}^{2}}{1/\sigma_{1}^{2} + 1/\sigma_{2}^{2}}$
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$\sigma_{m}^{2} = \frac{1}{1/\sigma_{1}^{2} + 1/\sigma_{2}^{2}} + \sigma_{s}^{2}$
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$\Rightarrow x_{m} = 4.21 \pm 3.12$
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### Combination of Measurements II
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1)
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$x_{1} = 318$
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$x_{2} = 335$
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For counting experiments assume Poisson Distribution:
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$x_{m} = \frac{x_{1}+ x_{2}}{2}$
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$\sigma_{m}^{2} = x_{m}$
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$\Rightarrow x_{m} = 326.5 \pm 12.8$
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### Comparison of Measurements
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1)
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$A = \frac{N_{1}-N_{2}}{N_{1}+N_{2}}$
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$\frac{\delta A}{\delta N_{1}} = \frac{1}{N_{1}+N_{2}}-\frac{N_{1}-N_{2}}{(N_{1}+N_{2})^{2}} = \frac{N_{1}+N_{2}-(N_{1}-N_{2})}{(N_{1}+N_{2})^{2}} = \frac{N_{1}+N_{2}}{(N_{1}+N_{2})^{2}} = \frac{1}{N}$
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$\frac{\delta A}{\delta N_{2}} = -\frac{1}{N}$
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For counting experiments assume Poisson Distribution:
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$\sigma_{N_{1}}^{2} = \sigma_{N_{2}}^{2} = \frac{N}{2}$
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$\sigma_{A}^{2} = \left( \frac{\delta A}{\delta N_{1}} \right)^{2} \sigma_{N_{1}}^{2} + \left( \frac{\delta A}{\delta N_{2}} \right)^{2} \sigma_{N_{2}}^{2} = \frac{1}{N}$
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$\Rightarrow \sigma_{A} = \frac{1}{\sqrt{N}}$
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2)
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$\text{cov}(N,N') = \langle N N' \rangle - \langle N \rangle\langle N'\rangle = \langle ( N' + \delta N ) N' \rangle + \langle N' - \delta N \rangle \langle N'\rangle = \langle N'^{2} \rangle + \langle \delta N \rangle \langle N' \rangle - \langle N' \rangle^{2} - \langle \delta N \rangle \langle N' \rangle = \langle N'^{2} \rangle - \langle N' \rangle^{2} \approx N'$
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$\rho_{N,N'} = \frac{\text{cov}(N,N')}{\sqrt{N}\sqrt{N'}} = \frac{N'}{\sqrt{N N'}} = \sqrt{\frac{N'}{N}}$
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3)
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$\left( \frac{\delta \vec{A}}{\delta \vec{N}} \right) =
\left(\begin{array}{cccc}
\frac{\delta A}{\delta N_{1}} & \frac{\delta A}{\delta N_{2}} & \frac{\delta A}{\delta N'_{1}} & \frac{\delta A}{\delta N'_{2}} \\
\frac{\delta A'}{\delta N_{1}} & \frac{\delta A'}{\delta N_{2}} & \frac{\delta A'}{\delta N'_{1}} & \frac{\delta A'}{\delta N'_{2}}
\end{array}\right) =
\left(\begin{array}{cccc}
\frac{1}{N} & -\frac{1}{N} & 0 & 0 \\
0 & 0 & \frac{1}{N'} & -\frac{1}{N'}
\end{array}\right)$
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$\text{cov}(A,A') = \left(\begin{array}{cccc}
\frac{1}{N} & -\frac{1}{N} & 0 & 0 \\
0 & 0 & \frac{1}{N'} & -\frac{1}{N'}
\end{array}\right)
\left(\begin{array}{cccc}
N_{1} & 0 & N'_{1} & 0 \\
0 & N_{2} & 0 & N'_{2} \\
N'_{1} & 0 & N'_{1} & 0 \\
0 & N'_{2} & 0 & N'_{2}
\end{array}\right)
\left(\begin{array}{cc}
\frac{1}{N} & 0 \\
-\frac{1}{N} & 0 \\
0 & \frac{1}{N'} \\
0 & -\frac{1}{N'}
\end{array}\right) = \left(\begin{array}{cc}
\frac{1}{N} & \frac{1}{N} \\
\frac{1}{N} & \frac{1}{N'}
\end{array}\right)$
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$\Rightarrow \rho_{A,A'} = \frac{\text{cov}(A,A')}{\sigma_{A}\sigma_{A'}} = \frac{1/N}{\sqrt{1/N}\sqrt{1/N'}} = \sqrt{\frac{N'}{N}}$
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4)
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$\sigma_{A-A'}^{2}= \sigma_{A}^{2} + \sigma_{A'}^{2} - 2 \text{cov}(A,A') = \frac{1}{N} + \frac{1}{N'} - \frac{2}{N} = \frac{1}{N'} - \frac{1}{N} = \sigma_{A'}^{2} - \sigma_{A}^{2}$
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### Combination of Measurements I
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1)
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$x_{1} = 12 \pm 3 \text{ (sta.)} \pm 2 \text{ (sys.)}$
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$x_{2} = 8 \pm 4 \text{ (sta.)} \pm 2 \text{ (sys.)}$
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For derivation please watch lecture slide 127.
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$x_{m} = \frac{x_{1}/\sigma_{1}^{2} + x_{2}/\sigma_{2}^{2}}{1/\sigma_{1}^{2} + 1/\sigma_{2}^{2}}$
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$\sigma_{m}^{2} = \frac{1}{1/\sigma_{1}^{2} + 1/\sigma_{2}^{2}} + \sigma_{s}^{2}$
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$\Rightarrow x_{m} = 10.56 \pm 3.12$
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### Combination of Measurements II
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1)
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$x_{1} = 318$
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$x_{2} = 335$
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For counting experiments assume Poisson Distribution:
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$x_{m} = \frac{x_{1}+ x_{2}}{2}$
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$\sigma_{m}^{2} = x_{m}$
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$\Rightarrow x_{m} = 326.5 \pm 12.8$
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### Comparison of Measurements
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1)
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$A = \frac{N_{1}-N_{2}}{N_{1}+N_{2}}$
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$\frac{\delta A}{\delta N_{1}} = \frac{1}{N_{1}+N_{2}}-\frac{N_{1}-N_{2}}{(N_{1}+N_{2})^{2}} = \frac{N_{1}+N_{2}-(N_{1}-N_{2})}{(N_{1}+N_{2})^{2}} = \frac{N_{1}+N_{2}}{(N_{1}+N_{2})^{2}} = \frac{1}{N}$
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$\frac{\delta A}{\delta N_{2}} = -\frac{1}{N}$
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For counting experiments assume Poisson Distribution:
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$\sigma_{N_{1}}^{2} = \sigma_{N_{2}}^{2} = \frac{N}{2}$
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$\sigma_{A}^{2} = \left( \frac{\delta A}{\delta N_{1}} \right)^{2} \sigma_{N_{1}}^{2} + \left( \frac{\delta A}{\delta N_{2}} \right)^{2} \sigma_{N_{2}}^{2} = \frac{1}{N}$
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$\Rightarrow \sigma_{A} = \frac{1}{\sqrt{N}}$
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2)
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$\text{cov}(N,N') = \langle N N' \rangle - \langle N \rangle\langle N'\rangle = \langle ( N' + \delta N ) N' \rangle + \langle N' - \delta N \rangle \langle N'\rangle = \langle N'^{2} \rangle + \langle \delta N \rangle \langle N' \rangle - \langle N' \rangle^{2} - \langle \delta N \rangle \langle N' \rangle = \langle N'^{2} \rangle - \langle N' \rangle^{2} \approx N'$
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$\rho_{N,N'} = \frac{\text{cov}(N,N')}{\sqrt{N}\sqrt{N'}} = \frac{N'}{\sqrt{N N'}} = \sqrt{\frac{N'}{N}}$
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3)
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$\left( \frac{\delta \vec{A}}{\delta \vec{N}} \right) =
\left(\begin{array}{cccc}
\frac{\delta A}{\delta N_{1}} & \frac{\delta A}{\delta N_{2}} & \frac{\delta A}{\delta N'_{1}} & \frac{\delta A}{\delta N'_{2}} \\
\frac{\delta A'}{\delta N_{1}} & \frac{\delta A'}{\delta N_{2}} & \frac{\delta A'}{\delta N'_{1}} & \frac{\delta A'}{\delta N'_{2}}
\end{array}\right) =
\left(\begin{array}{cccc}
\frac{1}{N} & -\frac{1}{N} & 0 & 0 \\
0 & 0 & \frac{1}{N'} & -\frac{1}{N'}
\end{array}\right)$
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$\text{cov}(A,A') = \left(\begin{array}{cccc}
\frac{1}{N} & -\frac{1}{N} & 0 & 0 \\
0 & 0 & \frac{1}{N'} & -\frac{1}{N'}
\end{array}\right)
\left(\begin{array}{cccc}
N_{1} & 0 & N'_{1} & 0 \\
0 & N_{2} & 0 & N'_{2} \\
N'_{1} & 0 & N'_{1} & 0 \\
0 & N'_{2} & 0 & N'_{2}
\end{array}\right)
\left(\begin{array}{cc}
\frac{1}{N} & 0 \\
-\frac{1}{N} & 0 \\
0 & \frac{1}{N'} \\
0 & -\frac{1}{N'}
\end{array}\right) = \left(\begin{array}{cc}
\frac{1}{N} & \frac{1}{N} \\