# 2021-06-22

## 12:47

mapping cone as a pushout and mapping fiber as pullback : • $$M_f\to X\to Y$$ yields a LES in homotopy $$\pi_* M_f \to \pi_* X \to \pi_* Y$$.
• $$X\to Y\to C_f$$ yields $$\pi_*(Y, X) = \pi_{*-1} C_f$$?
• This should be an easy consequence of the LES in homotopy.

## Probability Review • Binomial and Poisson distributions tend to normal distributions?
• Normal distribution rule: 68, 95, 99.7.
• \begin{align*} {\mathbb{E}}[r(x)] \coloneqq\int r(x) f(x) \,dx && \text{or } \sum_k r(k) P(x=k) .\end{align*}
• $${\mathsf{Var}}(x) = E[x^2] - E^2[x]$$
• Marginal densities: if $$f(x, y)$$ is a distribution, \begin{align*} f_x(x) \coloneqq\int f(x, y) \,dy && P(x\times y \in A) = \iint_A f(x, y) \,dx\,dy .\end{align*}
• Independent if $$f(x, y) = f_x(x) f_y(y)$$.
• If $$x$$ has density $$f_x$$ and $$y \coloneqq r(x)$$ for a continuous increasing function, setting $$s \coloneqq r^{-1}$$, $$y$$ has density \begin{align*} g_y(y) = f_x(s(y)) s'(y) .\end{align*}
• Given a failure rate $$\lambda$$ in failures over time and a time span $$t$$, the probability of failing $$k$$ times in $$t$$ units of time is $$\operatorname{Poisson}(\lambda t)$$.
• The time between failures is $$\operatorname{Exponential}(\lambda)$$.

## Combinatorics Review

• Prop: the number of integer compositions of $$n$$ into exactly $$k$$ parts is $${n-1 \choose k-1}$$, so $$\sum a_i = n$$ with $$a_i\geq 1$$ for all $$i$$.

• Lay out $$n$$ copies of 1 with $$n-1$$ blanks between them. Choose to put a plus sign or a comma in each slot: choose $$k-1$$ commas to produce $$k$$ blocks.
• Prop: the number of compositions of $$n$$ into any number of parts is $$2^{n-1}$$.

• Sum over compositions with exactly $$k$$ parts and use the identity $$\sum_{k=1}^{n-1} {n-1 \choose k-1} = 2^{n-1}$$.
• Prop: the number of weak compositions of $$n$$ into exactly $$k$$ parts is $${n+k-1 \choose k-1}$$.

• Add one to each piece to get a (strong) composition of $$n+k$$ into $$k$$ parts.
• Quotient set of compositions by permutations to get integer partitions.

• Stirling numbers : the number of ways to partition an $$n$$ element set into exactly $$k$$ nonempty unlabeled disjoint blocks whose disjoint union is the original set.

• Prop: recurrence for partitions : \begin{align*}s(n+1, k) = s(n, k-1) + ks(n, k).\end{align*}
• Proof: check if a given partition of $$n+1$$ into $$k$$ parts contains the singleton $$\left\{{n+1}\right\}$$.
• If so, delete and count partitions of $$n$$ into $$k-1$$ parts.
• If not, add $$\left\{{n+1}\right\}$$ to any of the $$k$$ parts.