Axioms of Probability. Axioms for probability: 1.P(E) 0 for any eventE ≥ 2.P(S)=1
3. IfE ,E ,E ,... are disjoint events, thenP(E E E ...)= ∞ P(E ) 1 2 3 1 ∪ 2 ∪ 3 ∪ i=1 i Thoerem (Basic theorems of probability). LetA andB be events. � 1.P( ) = 0 ∅ 2. IfA B, thenP(A) P(B) ⊆ ≤ 3.P(A) = 1 P(A C ) − 4.P(A) =P(A B)+P(A B C ) ∩ ∩ 5. P(A B)=P(A) +P(B) P(A B) ∪ − ∩ Method. The number of ways to selectk elements from ann-element set is...
Order matters Order doesn’t matter n+k 1 With replacement nk − k � � n! n n! Without replacement = (n k)! k (n k)!k! − � � − Definition. LetA andB be events withP(B)= 0. The conditional probability ofA givenB is � P(A B) P(A B)= ∩ | P(B) Definition. EventsA andB are independent if and only ifP(A B)=P(A)P(B). ∩ Thoerem (Multiplication rule for probabilities). IfP(B)= 0, thenP(A B)=P(A B)P(B). � ∩ | Thoerem (The Law of Total Probability). If eventB has probability strictly between0 and1, then for any eventA,P(A) =P(A B)P(B)+P(A B C )P(B C ). | | P(A B)P(B) Thoerem (Bayes’ Law). IfA andB are events with positive probability, thenP(B A) = | | P(A) Definition.A random variableX assigns a number to each outcome in the sample spaceS. 1. All random variables have a cumulative distribution function (CDF):F(x) =P(X x). ≤ 2. A discrete random variable has a probability mass function (PMF):m(x) =P(X=x). Definition. The total number of successes inn independent, identically distributed (iid) Bernoulli trials with parameterp is a random variable with a binomial distribution. The PMF of a random variableX having a
n x n x binomial distribution with parametersn andp is b(x) = p (1 p) − forx=0,1, . . . , n x − � � Proposition. The mean of a binomial distribution isµ= np.
Definition. LetX 1,X2,... be a sequence of independent, identically distributed (iid) Bernoulli trials, all with probability of successp. LetN be the trial on which thefirst success occurs. The random variableN is said to n 1 have a geometric distribution with parameterp and its PMF is g(n) =p(1 p) − forn=1,2,3,... − 1 Proposition. The mean of a geometric distribution isµ= . p Definition. Supposen elements are to be selected without replacement from a population of sizeN of whichk are successes. The number of successes selected is a hypergeometric random variable and its PMF is k N k x n−x h(x) = N − for those integersx which make all terms positive. � ��n � � � nk Proposition. The mean of a hypergeometric distribution isµ= . N