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