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Set
collection of elements
Subset of a set
each element in A is an element of B (B may contain elements not in A, but A is totally contained in B)
Union
set of all elements in either A or B or both (AuB = A+B-AnB)
Intersection
set of all elements that are in both A and B
DeMorgan’s Laws
(i)(AuB)’ = A’nB’ (ii) (AnB)’ = A’uB’
Empty set
set that contains no elements (null set)
Disjoint set
intersection of A and B is the empty set
Power rule
f(x) = cxn f’(x) = cnxn-1
Product rule
f(x) = g(x)*h(x) f’(x) = g’(x)*h(x) + g(x)*h’(x)
Quotient rule
f(x)=g(x)/h(x) f’(x)=(h(x)*g’(x) – g(x)*h’(x))/(h(x)^2)
Chain rule
f(x)=g(h(x)) f’(x)=g’(h(x))*h’(x)
Derivative of f(x)=e^g(x)
f’(x)=g’(x)*e^g(x)
Derivative of f(x)=ln(g(x))
f’(x)=g’(x)/g(x)
Derivative of f(x)=a^x
f’(x)=a^x*lna
Derivative of f(x)=logbx
f’(x)=1/(xlnb)
L’Hospital’s rules
if f(x)=g(x)/h(x) and the limit of the two individual functions is either 0 or infinity then the limit of f(x) can be found by taking the derivative of the two functions until the limit can be determined
Partial differentiation
take the derivative of the function with respect to the specified variable keeping other variables constant
To find the integral of two functions being added
add the integrals of the functions
Integrate f(x) = a^x
F(x)=a^x/lna
Integrate f(x)= xe^(ax)
F(x)= (xe^ax)/a – (e^ax)/a^2
Integration when f(x) is infinite or not defined at an endpoint
take the limit as that endpoint approaches the discontinuity
Integrations if there is a point of discontinuity within the interval
separate into two integrals at the point of discontinuity
Geometric progression (ie a, ar, ar^2…)
a*(1-r^n)/(1-r)
Arithmetic progression (ie a, a+d, a+2d, a+3d,…)
na+d(n^2-n)/2
Special arithmetic (1,2,3,4…)
n(n+1)/2
Sample point
simple outcome of a random experiment
Sample space
collection of all possible sample points related to a specific experiment
Mutually exclusive outcomes
events that cannot occur simultaneously (disjoint outcomes)
Exhaustive outcomes
all outcomes combine to be the entire probability space
Event
any collection of sample points; subset of the probability space
Complement of an event
all sample points in the probability space that are not in the event
Subset of an event AcB
event B contains all the sample points in event A; occurrence of event A implies that event B occurs
Partition of event A
events c1, c2, c3…form a partition of event A if A is the union of all c events and the c events are mutually exclusive
Probability Propertites
(i) distributive property (ii) union of event and complement equals the sample space (iii) intersection of event and complement is the null set (iv) if AcB then AuB=B and AnB=A
Discrete probability space
set of finite or countable infinite number of sample points; P(a) denotes the probability that event a will occur
Probability conditions
(i) all probabilities are between [0,1] (ii) all of the probabilities sum to 1
Uniform probability function
a probability space has a finite number of sample points and each sample points has the same probability of occurring
Continuous probability space
outcomes which can be any real number in some interval
Conditional probability P(B|A)
probability of B given A; if P(A) >0 the P(B|A)=P(BnA)/P(A)
Law of total probability
P(B)=P(B|A)*P(A) + P(B|A’)*P(A’)
Bayes Thm
P(A|B)=P(AnB)/P(B)=P(B|A)*P(A)/P(B)
Bayes Thm with a partition
P(Aj|B)=P(BnAj)/P(B)=P(BnAj)/ΣP(BnAi)=P(B|Aj)*P(Aj)/ΣP(B|Ai)*P(Ai)
Independent events
events are independent if P(AnB)=P(A)*P(B)
Conditional probability and independence
P(AnB)=P(B|A)*P(A)=P(A|B)*P(B)
Conditional probability properties
(i) P(A’|B)=1-P(A|B) (ii) P(AuB|C)=P(A|C)+P(B|C)-P(AnB|C) (iii) if AcB then P(A|B)=P(A)/P(B) (iv) if A and B are independent then their complements are also independent
n!
number of ways n distinct objects can be ordered
nPk
choosing an ordered subset of size k without replacement; nPk=n!/(n-k)!
nCk
number of ways of choosing a subset of size k<=n without replacement and without regard to order in which the objects are chosen; nCk=n!/(k!*(n-k)!); also called binomial coefficient
multinomial theorem
given n objects of which n1 are type 1, n2 are type 2…in order to find number of ways of choosing a subset of size k where there are k1 objects that are type 1… N!/(k1!*...*kt!)
random variable
a function on a probability space it assigns a real number x(s) to each sample point s in the sample space; possible values that can occur and the probabilities of these values occurring
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