Week 1: 18-20 Aug (asynchronous)
□ Read on random variables, pdfs, characteristic functions, convolutions, joint pdfs, conditional pdfs.
Week 2: 25-27 Aug: Stochastic variables
References vK I, RL 4
□ Provide a scientific or mathematical definition of the probability of an event/experimental outcome.
□ Provide a technical description of a stochastic variable.
□ Give the conditions needed for a function to be a valid probability distribution function.
□ Demonstrate how the characteristic function is used as a generating function for the cumulants of a probability distribution.
□ Define the elements of the covariance matrix.
□ Give the central limit theorem and emphasize the conditions needed for it to hold.
W03 4 Sep: Averages
References vK I
! Moments and cumulants are expectation values of polynomial functions of a random variable. They provide measurable information about unknown probability distributions.
□ Give the characteristic function as an expectation value of a function of a stochastic variable.
□ Demonstrate how the characteristic function and its logarithm generate moments and cumulants (expectation value of special linear combinations of powers of the stochastic variable).
□ Discuss some interesting and useful situations where "special linear combinations of powers of a stochastic variable" are needed.
W04 8-10 Sep: Multivariate distributions
References vK I.3
□ Define the notation used for a multivariate PDF. Show how these are normalized.
□ Define the notation used for a reduced or marginal multivariate PDF as a partially summed multivariate PDF. Try doing the same for quantum density matrices with a partial trace and highlight the similarities.
□ Define the notation used for a multivariate conditional PDF. Write down Bayes' rule as an equation and in words.
W05 16-18 Sep: Functions of stochastic variables
References vK I.4, I.5
□ Show how the addition of two independent stochastic variables leads to a convolution of PDFs.
□ Show that the addition of two independent stochastic variables is simple in Fourier space. Discuss if this simplification can be done if the variables are dependent.
□ Discuss some applications why one would need transformations or functions of a stochastic variable like Y = f(X). Write down the characteristic function of Y and apply it to the addition of squares of a stochastic variable (e.g. Maxwell's speed distribution as an energy density, or the similar construction of a χ2 distribution for error analysis).
For reference, past course guide is here: Physics 151
Faculty
- Francis Paraan