STAT 230 - Chapter 1

• Classical Definition - $\frac{\text{number of ways the event can occur}}{\text{number of outcomes in S}}$
• Relative Frequency - Probability of an event is proportion of times the event occurs in a long series of repetitions of an experiment / process.
  • E.g probability of getting a 2 from a rolled dice is $\frac{1}{6}$
• Subjective Probability - Probability of an event is a measure of how sure person making the statement is that the event will happen
• Discrete - Consists of a finite or countably infinite set of sample points
• Simple definition
  • $0\le P(a_i)\le 1$
  • ${\sum }_{\text{all i}}P(a_i)={\sum }_{\text{all }{a}_i}P(a_i)=1$
  • Let $S=\{a_1,a_2,a_3,\dots \}$ be a discrete sample space
• Odds in favor of an event A is the probability the event occurs divided by the probability it does not occur, or simply $\frac{P(A)}{1-P(A)}$, odds against the event is reciprocal of this or simply $\frac{1-P(A)}{P(A)}$
• Additional Rule - Do job 1 in p ways and job 2 in q ways, then we can do either job 1 OR job 2 (but not both) in p + q ways
• Multiplication Rule Do job 1 in p ways and for each of these ways we can do job 2 in q ways. Then we can do both job 1 AND job 2 in p x q ways
• $n\times (n-1)\times \cdots \times 1$ ordered arrangements of length n using each symbol once and only once. This is denoted by $n!$
• Permutation - $n\times (n-1)\times \cdots \times (n-k+1)$ ordered arrangements of length $k\le n$ each symbol at most once. This product is denoted by $n^{(k)}$ (read “n to k factors”). Note that $n^{(k)}=\frac{n!}{(n-k)!}$ (ORDER DOES MATTER)
• Combination - $\frac{n!}{k!(n-k)!}$ (ORDER DOES NOT MATTER)
• Inclusion Exclusion Principle - $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
  • 3 events: $P(A\cup B\cup C)=P(A)+P(B)+P(C)-[P(A\cap B)+P(A\cap C)+P(B\cap C)]+P(A\cap B\cap C)$
• Conditional Probability - A and B be two events such that $P(B)>0$ then
  • $P(A|B)=\frac{P(A\cap B)}{P(B)}$
• Independence - $P(A\cap B)=P(A)\cdot P(B)$
• Law of Total Probability - $S={\sum }_{k=1}^n{B}_k$ where $B_i\cap B_j=\varnothing$ if $i\ne j$ then for any event, A, $P(A)={\sum }_{R=1}^{n}P(A|B_R)\cdot P(B_R)$
• Probability Mass Function - Mapping each outcome to the probability of it happening
• Cumulative Distribution Function - Cumulatively adding up the outcomes of the previous event
• Binomial Distribution - n independent trials with 2 possible outcomes
  • success: probability p
  • failure: probability 1 - p
  • X = # of successes in n trials then X follows a Binomial Distribution with params n and p
  • X ~ Bin(n, p)
• Geometric Distribution - X is number of trials required to observe first success in a sequence of experiments
  • X ~ Geo(p) then E(X) = $\frac{1}{p}$ and Var(X) = $\frac{(1-p)}{p^2}$
• Negative Binomial Distribution - X ~ NB(r, p)
  • X is the number of trials required to observe the $r^{th}$ success
  • $E(X)=\frac{r}{p}$
  • $Var(X)=\frac{r(1-p)}{p^2}$
• Poisson Distribution - X ~ Po($\lambda$)
  • P(X = x) = f(x) = $\frac{e^{-\lambda }{\lambda }^x}{x!}$, $x=0,1,2,\dots ,\text{ and }\lambda >0$
  • $\lambda$ - rate / mean “average # of successes”
  • $E(X)=\lambda =\mu$
  • $Var(X)=\lambda$