Math 480, Fall 2016, Assignment 12
From cartan.math.umb.edu
Algebra begins with the unknown and ends with the unknowable.
- - Anonymous
Carefully define the following terms, then give one example and one non-example of each:
- Convex function.
Carefully state the following theorems (you do not need to prove them):
- Jensen's inequality.
- Log sum inequality.
- Theorem concerning the entropy of an amalgamation.
Solve the following problems:
- Let $(S,p)$ be a finite probability space modelling the simultaneous roll of two distinguishable fair six-sided dice. Let $X$ be the random variable "sum of the two dice." Explicitly compute the system of events $\mathscr{X}$ corresponding to $X$ (i.e. for each possible value of $X$, list all outcomes corresponding to that value). Next, let $Y=(X-7)^2$. Compute the system of events $\mathscr{Y}$ corresponding to $Y$. Finally, compute $H(\mathscr{X})$ and $H(\mathscr{Y})$.
- (Functions of a random variable as amalgamations). Let $(S,p)$ be a finite probability space, let $A$ be a finite set, let $X$ be an $A$-valued random variable on $S$, let $B$ be another finite set, let $f:A\rightarrow B$ be any function, and let $Y$ denote the composition $f\circ X$, so that $Y$ is a $B$-valued random variable on $S$. Finally, let $\mathscr{X}$ be the system of events corresponding to $X$ (i.e. the events in $\mathscr{X}$ are those of the form "$X$ takes the value $a$" for $a\in A$) and let $\mathscr{Y}$ denote the system of events corresponding to $Y$. Show that $\mathscr{Y}$ is an amalgamation of $\mathscr{X}$.
