S2 summary

Chapter 1 summary - The binomial and Poisson distributions

1. Binomial distribution
   
   X \sim B(n,p)
   
   
   \displaystyle {\rm P}(X=r) = {n\choose r}p^r(1-p)^{n-r} \qquad r = 0, 1, ... n
   
   \mu = {\rm E}(X) = np
   
   \sigma^2 = {\rm Var}(X) = np(1-p) = npq
   
   
2. Poisson distribution
   
   X \sim {\rm Po}(\lambda)
   
   
   \displaystyle {\rm P}(X=r) = {e^{-\lambda}\lambda^r\over r!}
   
   \mu = {\rm E}(X) = \lambda
   
   \sigma^2 = {\rm Var}(X) = \lambda
   
   
3. Poisson approximation to binomial
   
   X \sim {\rm B}(n,p)
   
   
   if n is large
   and p is small
   then X \approx \sim {\rm Po}(np)

Chapter 2 summary - Continuous random variables

1. Continuous random variable X
   
   \displaystyle \int_{-\infty}^{\infty} {\rm f}(x){\rm d}x = 1
   
   \displaystyle \mu = {\rm E}(X) = \int_{-\infty}^{\infty} x{\rm f}(x){\rm d}x
   
   \displaystyle \sigma^2 = {\rm E}(X^2) - \mu^2 = \int_{-\infty}^{\infty} x^2{\rm f}(x){\rm d}x - \mu^2
   
   
2. Cumulative distribution function {\rm F}(x)
   
   0 \leq {\rm F}(X) \leq 1
   
   \displaystyle {\rm F}(x_0) = {\rm P}(X \leq x_0) = \int_{-\infty}^{x_0}{\rm f}(x){\rm d}x
   
   Median m satisfies {\rm F}(m) = 0.5
   Quartile Q_1 satisfies {\rm F}(Q_1) = 0.25
   Quartile Q_3 satisfies {\rm F}(Q_3) = 0.75

Chapter 3 summary - Continuous distributions

1. A random variable X, having a continuous uniform distribution over the interval (\alpha, \beta) has p.d.f.
   
   \displaystyle {\rm f}(x) = \cases{{1\over \beta - \alpha}, &\alpha < x < \beta \cr 0, &\text {otherwise.}}
   
   
2. For a random variable X, having a uniform distribution
   
   \displaystyle {\rm E}(X) = {\alpha + \beta\over 2}
   
   {\rm Var}(X) = {1\over 12}(\beta - \alpha)^2
   
   
3. A random variable X \sim {\rm B}(n,p) can be approximated by Y \sim {\rm N}(\mu, \sigma^2) when \mu = np and \sigma^2 = np(1-p) = npq provided that n is large, np > 5 and n(1-p) = nq > 5.
   
   
4. A random variable X \sim {\rm Po}(\lambda) can be approximated by
   
   Y \sim {\rm N}(\lambda, \lambda) \qquad for \lambda > 10

Chapter 4 summary - Hypothesis tests

1. A population is a collection of individual items.
   
   
2. A sample is a selection of individual members or items from a population.
   
   
3. A finite population is one in which each individual member can be given a number.
   
   
4. An infinite population is one in which it is impossible to number each member.
   
   
5. A countably infinite population is one which is infinite in size, but each member can be given an individual number.
   
   
6. A sampling unit is an individual member of a population.
   
   
7. A sampling frame is a list of sampling units used in practice to represent a population. In some instances the two will be identical, in others the sampling frame will represent the population as accurately as possible.
   
   
8. In practice a sample is a collection of sampling units drawn from a sampling frame.
   
   
9. A hypothesis test is a mathematical procedure to examine a value of a population parameter proposed by the null hypothesis {\rm H}_0 compared with an alternative hypothesis {\rm H}_1.
   
   
10. The critical region is the range of values of a test statistic T that would lead you to reject {\rm H}_0.