The theory of distribution tries to remedy this by imbedding classical functions in a larger class of objects, the so called distributions (or general functions).

In the previous lecture we defined a ∂2 n-distribution with n degrees of freedom as a distribution of the sum 2 X1 + . . . + Xn , 2 where Xis are i.i.d. standard normal.

For instance, to solve a PDE such as Lu = f where L is a constant-coefficient differential operator, and f is a given test function, one can often use distributions to obtain a (smooth) solution of the form u = f …

The Normal Distribution Based on a chapter by Chris Piech the normal (a.k.a. Gaussian) random variable, parametrized by a mean ( ) and variance ( 2). The normal is important for many reasons: it …

A Guide to Distribution Theory and Fourier Transforms [2], by Robert Strichartz. The discussion of distributions in this book is quite compre-hensive, and at roughly the same level of rigor as this course.

The exponential distribution can be used to model lifetimes, anal-ogous to the use of the geometric distribution in the discrete case. In fact, the exponential distribution shares the “memoryless” property …

The Bernoulli distribution, named after the swiss mathematician Jacques Bernoulli (1654– 1705), describes a probabilistic experiment where a trial has two possible outcomes, a success or a failure.