Approximations:
- Binomial => Poisson | n is large and p is small | Ideal case: n>50 and p < 0.5 | $\lambda = np$
- Binomial => Normal | np, nq > 10 , n large and fixed p | $\mu = np$ $\sigma ^2 = npq $
Nuances:
1. Exponential:
- ~Continuous version of the Geometric distribution. Models the time elapsed until one random event occurs.
- $ f_X(x) = \lambda_1 e^{-\lambda_1x}$ and $f_Y(y) = \lambda_2 e^{-\lambda_2x}$ then $P(X<y) ==”” \<span=”” class=”hiddenSpellError” pre=”” data-mce-bogus=”1″>frac {\lambda_1}{\lambda_1 + \lambda_2}$
- $E[X^k] = k! \beta^k$
- For a positive deductible ‘d’,
- $E[X/X>d] = \beta + d$ and
- $E[X-d/X>d] =\beta$
2. Weibull :
- $E[X^k] = \theta ^n\Gamma (1+\frac{k}{\beta })$k = 1,2,3…
- $E[X^k] = \theta ^n\Gamma (1+\frac{k}{\beta })$ k = 1,2,3…
3. Uniform :
- $E[X^k] = \frac{b^{n+1} – a^{n+1} }{(n+1)(b-a)}$