I saw the following index shift involving the Taylor Series for $\sin(x)$. $$\sin(x)=\sum^{\infty}_{n=1}(-1)^{n-1}\frac{x^{2n-1}}{(2n-1)!}=\sum^\infty_{n=0}(-1)^{n}\frac{x^{2n+1}}{(2n+1)!}$$ This second sum seems right to me, but the index shift seems funny. If I shifted the index down one - from...

I find it hard to notice when do I have a Multinomial distribution and if its possible to "transform" problems into a Multinomial distribution problems. For example I have the following exercise: $15$ people come for a test $21$ quizzes has been printed $7$ quizzes are type A ...

In Multinomial Distribution, we have \begin{align} f(x_1,\ldots,x_k;n,p_1,\ldots,p_k) & {} = \Pr(X_1 = x_1\mbox{ and }\dots\mbox{ and }X_k = x_k) \\ \\ & {} = \begin{cases} { \displaystyle {n! \over x_1!\cdots x_k!}p_1^{x_1}\cdots p_k^{x_k}}, \quad & \mbox{when } \sum_{i=1}^k x_i=n \\ \\ 0 & ...

I was reading the answer of this question : Convergence of measure sequences bounded by a finite measure But I did not understand some parts in the proof of $\mu: \mathcal{A}\longrightarrow[0,\infty]$. First: I did not understand why the $\epsilon$ in the definition of the limit is the same...