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.

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.


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.


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?


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.