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Some Mathematical Preliminaries

Definition of exponentiation

\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
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
f(a+h) = f(a) + f'(a)h + \frac{1}{2} f”(a) h^2 + \frac{1}{3!} f”'(a) h^3 + \ldots

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\\
\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}

\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

Lagrange multipliers


Gamma Function