Download Final Cheat Sheet for Finite Probability and Applications | MT 004 and more Study notes Mathematics in PDF only on Docsity! MT004 Final Cheat Sheet İlker S. Yüce December 14, 2009 Chapter 5 Inclusion-Exclusion Principle: n(S ∪ T ) = n(S) + n(T )− n(S ∩ T ). De Morgan’s Law: (S ∩ T )′ = S′ ∪ T ′, (S ∪ T )′ = S′ ∩ T ′. Permutations: P (n, r) = n(n− 1)(n− 2) · · · (n− r + 1). Combinations: C(n, r) = P (n, r) r! = n! r!(n− r)! . Number of Ordered Partitions of Type (n1, n2, . . . , nm) where n = n1 + n2 + · · ·+ nm: n! n1!n2! . . . nm! . Number of Unordered Partitions of Type (r, r, . . . , r) where n = m× r: 1 m! · n! (r!)m . Chapter 6 Fundamental Property 1: Each of the probabilities p1, p2, . . . , pn is between 0 and 1. Fundamental Property 2: p1 + p2 + · · ·+ pN = 1. Equally Likely Events: Let S be a sample space con- sisting of N equally likely outcomes. Let E be any event. Then P (E) = number of outcomes in E N . Complement Rule: Let E be any event and E′ its complement. Then P (E) = 1− P (E′). Addition Principle: P (E) = P (s) + P (t) + · · ·+ P (z) where E = {s} ∪ {t} ∪ · · · ∪ {z}. Inclusion-Exclusion Principle: P (E ∪ F ) = P (E) + P (F )− P (E ∩ F ). Mutually Exclusive Events: P (E ∪ F ) = P (E) + P (F ) where E and F are mutually exclusive events, i.e., E ∩ F = ∅. Conditional Probability for Equally likely Outcomes: P (E|F ) = n(E ∩ F ) n(F ) = P (E ∩ F ) P (F ) . Product Rule: If P (F ) 6= 0, then P (E ∩ F ) = P (F )× P (E|F ). Independent Events: If E and F are independent events with nonzero probabilities, then P (E|F ) = P (E), P (F |E) = P (F ). Bayes’ Theorem: If B1, B2,. . . ,Bn are mutually ex- clusive events, and if B1∪B2∪ · · ·∪Bn = S, then for any event A in S, P (Bk|A) = P (Bk)P (A|Bk) P (B1)P (A|B1) + · · ·+ P (Bn)P (A|Bn) for k = 1, 2, . . . , n. Chapter 7 Binomial Probabilities: IfX is the number of ”suc- cesses” in n independent trials, where in each trial the probability of a ”successes” is p, then P (X = k) = C(n, k)pkqn−k for k = 0, 1, 2, . . . , n where q = 1− p. Sample Mean: If x1, x2, . . . , xn is a sample of n num- bers then x̄ = x1 + x2 + · · ·+ xn n . Sample Mean: If x1, x2, . . . , xn is a sample of n num- bers where the frequency of x1 is f1, the frequency of x2 is f2, and so forth, where f1 + f2 + · · · + fr = n then x̄ = x1f1 + x2f2 + · · ·+ xrfr n = x1 f1 n + · · ·+ xr fr n . Population Variance: If x1, x2, . . . , xn is a sample of n numbers where the frequency of x1 is f1, the fre- quency of x2 is f2, and so forth, where f1 +f2 + · · ·+ fr = N then σ2 = (x1 − µ)2f1 + · · ·+ (xr − µ)2fr N 1
