Recent content by donutmax

Is the following correct? \frac{d}{dt}\sum_{n=0}^{\infty}\frac{2^{n}t^{n}}{(n+1)!}=\sum_{n=0}^{\infty}\frac{d}{dt}\frac{2^{n}t^{n}}{(n+1)!}

\sum_{n=0}^{\infty} \frac{2^{n+1}(n+1)t^n}{(n+2)!}=\sum_{n=0}^{\infty} \frac{2^{n}nt^{n-1}}{(n+1)!} How are the above summation equal?

M(t)=E(e^{ty})=\sum_{y=0}^{n} e^{ty}p(y) Is this correct?

Correction: K(t) is: K(t)=K_1t+K_2\frac{t^2}{2!}+... where K_n=K^{(n)}(0)

Cumulative generating function is K(t)=K_1(t)t+K_2(t)\frac{t^2}{2!}+K_3(t)\frac{t^3}{3!}+... where K_{n}(t)=K^{(n)}(t) Now K(t)=ln M(t)=ln E(e^{ty})=ln E(f(0)+f'(0)\frac {t}{1!}+f''(0)\frac{t^2}{2!}+...)=ln E(1+\frac{t}{1!}y+\frac{t^2}{2!} y^2+...)=ln [1+\frac{t}{1!} E(Y)+\frac{t^2}{2!}...

Is the following correct? M(t)=1+t\mu'_1+\frac{t^2}{2!}\mu'_2+\frac{t^3}{3!}\mu'_3+... =\sum_{n=0}^{\infty} \frac {E(Y^n)t^n}{n!} where \mu'_n=E(Y^n)

hi! are the following power series equivalent? ln(1+x)=\sum_{n=0}^{\infty} \frac{(-1)^n n! x^{n+1}}{(n+1)!} =\sum_{n=0}^{\infty} \frac{(-1)^n x^{n+1}}{n+1}