User:Kworb/Sandbox



Y = \begin{cases} P(A\mid X=0) & \text{if }X=0; \\ P(A\mid X=1) & \text{if }X=1. \end{cases} $$



f(x) = \begin{cases} \beta_2 \cdot \log(1+|x-\beta_1|) & \text{if }x>\beta_1; \\ -\beta_2 \cdot \log(1+|x-\beta_1|) & \text{if }x<-\beta_1; \\ 0 & \text{if }-\beta_1 \leq x \leq \beta_1. \end{cases} $$



\Delta IM = \Delta IM_{Y-1} + f(TGR) + g(CTGR) + h(TR - TTR) \, $$


 * $$(x+y)^n=\sum_{k=0}^n{n \choose k}x^{n-k}y^{k}$$

So


 * $$(1+2x)^n=\sum_{k=0}^n{n \choose k}1^{n-k}2^{k}x^{k}=\sum_{k=0}^n{n \choose k}2^{k}x^{k}$$

Coefficient of x^2:


 * $$2^{2}{n \choose 2}$$

Coefficient of x:


 * $$2{n \choose 1}=2n$$

So


 * $$2 \cdot 2n=4{n \choose 2}$$


 * $$n={n \choose 2}$$

Or


 * $$n=\frac{n!}{2!(n-2)!}=\frac{n!}{2(n-2)!}$$

Or


 * $$2n=\frac{n!}{(n-2)!}=n(n-1)=n^{2}-n$$

So


 * $$n^{2}-3n=0$$

And n=3