# PHYCS 3730/6720 Lab Exercise

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 $$\bar k$$.

#### 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?

#### 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.)