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PHYCS 3730/6720 Lab Exercise

Reading and references:
Answer file: **Mylab17.txt**
In this exercise we will be using Maple, rather than Python, because
we need to do some integrals.

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Exercise 1. Poisson distribution

Suppose a particle detector registers 5 counts. What is the
probability distribution for the mean value \(\bar k\)? Plot it
(use Maple). It is not a Gaussian. Hint: See Eq
(22) of the Introduction to Statistics. What is the mean of
\(\bar k\) for this probability distribution? What is the one
sigma confidence range for \(\bar k\), defined by the
requirement that 16% of the distribution lies above this range and 16%
below. Hint: Ask Maple to do the integral and just
play around with the integration range until you get something close
to 16% . The interval will not be symmetrical about the mean
value \(\bar k\).
Copy your Maple equation to the answer file. Give your answer for the
mean in the answer file. Give your answer for the confidence range in
the answer file.

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Exercise 2. Null result

Suppose a particle detector registers no counts. What is the
probability distribution for the mean value \(\bar k\)? Plot the
distribution. In this case one often speaks of a 95% confidence level
upper bound for \(\bar k\) such that only 5% of the time would we
expect to get zero counts with \(\bar k\) larger than this upper bound.
What is this upper bound?
Give your answer for the upper bound in the answer file.

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Exercise 3. Combined distribution

Suppose we measured 5 counts in one experiment and 0 counts in
another. Plot the probability distribution for \(\bar k\),
based on these two measurements. (It is the product of the
distributions in Exercises 1 and 2.) Find the most likely value of
\(\bar k\).
Give your answer for the most likely value in the answer file. Is it
what you expected?

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Exercise 4. Confidence level

This is the chisquare probability distribution function as discussed
in the Curve Fitting notes:
\[ P_{N}(\chi^{2}) = \frac{1}{2^{N/2}\Gamma(N/2)}e^{-\chi^{2}/2}(\chi^{2})^{(N/2)-1} \]
Plot it as a function of \( \chi^2 \) for \(N = 10\) degrees of
freedom.
Note that Maple understands the \( \Gamma \)
function as **GAMMA**. Calculate the probability of getting
a \( \chi^2 \) value greater than 16. (Integrate the function using
Maple.) Compare your result with the \( n_D = 10 \) curve in Figure 1
of the Curve Fitting notes. Compare it with the output of the course
utility **conf_level 16 10**. (Type this command in a terminal
window.)
Give your results and comparisons in the answer file.

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