Definition of exponentiation
\begin{eqnarray}
\lim_{n \rightarrow \infty}\left( 1+ \frac{1}{n} \right)^n =e\\
\lim_{n \rightarrow \infty}\left( 1+ \frac{x}{n} \right)^n =e^x
\end{eqnarray}
The second equation can easily be derived from the first. \\
This will be used in deriving the poisson distribution from the binomial distribution, and also in the proof of the central limit theorem.
Taylor Series
For $h$ small
\begin{eqnarray}
f(a+h) = f(a) + f'(a)h + \frac{1}{2} f”(a) h^2 + \frac{1}{3!} f”'(a) h^3 + \ldots
\end{eqnarray}
Leibniz’ rule for differentiation of integrals
Recall that cdf is the integral of pdf, while pdf is the derivative of cdf.\\
Leibniz rule is very useful in computing the function of a random variable\\
\begin{eqnarray}
\frac{d}{dt} \left(\int_{a(t)}^{b(t)} f(x,t)dx\right) = \int_{a(t)}^{b(t)} \frac{df(x,t)}{dt}dx + f(b(t),t) \frac{d b(t)}{dt}-f(a(t),t) \frac{d a(t)}{dt}
\end{eqnarray}
Example:
\begin{eqnarray}
\frac{d}{dt} \left(\int_{2t+3}^{t^2+5} (tx+4)dx\right) &=& \int_{2t+3}^{t^2+5} x dx + [t(t^2+5)+4]2t – [t(2t+3)+4]2 \nonumber\\
&=&\frac{(t^2+5)^2}{2} – \frac{(2t+3)^2}{2} + [t(t^2+5)+4]2t – [t(2t+3)+4]2 \nonumber
\end{eqnarray}
Lagrange multipliers