Problem 1

The Large Hadron Collider (and all colliders) operate by colliding bunches of ions together (a bunch at the LHC contains more than \(10^{11}\) protons). Each proton has a very small change of hitting another one so we can think of each potential collision as independent. Let’s say that for a given setup of the LHC the average number of collisions per bunch is \(\mu\).

Part 1 Write a function that determines the probability of more than one collision per bunch crossing in the case that there is at least one collision.

Part 2 Let’s assume that \(\mu = 40\). What is the probability that there are more than 60 collisions in a single bunch crossing?

Problem 2

In class, we discussed the error on the mean of a distribution via bootstrapping.

Part 1 Write a function that takes in an array of values and an order of a moment \(n\) and determines the error on that moment.

Part 2 Verify that the function works as expected for the mean by generating random samples according to a Poisson distribution of sizes 2, 5, 10, 50, 100, 500, 1000 and make a plot comparing the analytic uncertainty on the mean to that from the bootstrapping method. Make a plot (with labels) to show your results.

Problem 3

We looked at bootstrapping with 1D distributions in class. Generate a set of a 1000 random numbers according to a 2D Gaussian with a mean of 0 and \(\sigma\) in both dimension equal to 1. Let \(\rho\) be 0.95.

Use bootstrapping to find the 68% confidence interval for the difference of \(x\) and \(y\).

Use bootstrapping to find the 68% confidence interval on the Pearson correlation coefficient of \(x\) and \(y\).

Generate two new sets of random numbers with the same mean and widths but now with \(\rho\) of 0 and -0.95. Calculate the value of the Pearson correlation coefficient on this new dataset. Does what you see make sense?