Statistics, Sets, Counting, Probability, Estimation and Series Flashcards
average or (arithmetic) mean
the sum of the n numbers divided by n
median
order the numbers from least to greatest. If n is odd, the median is the middle number in the list. But if n is even, the median is the average of the two middle numbers.
mode
the number that occurs most often in a list.
statistics - range
The greatest value in data minus the least value
Standard deviation of n numbers
- Find their arithmetic mean
- Find the differences between the mean and each of the n numbers
- Square each difference
- Find the average of the squared differences
- Take the nonnegative square root of this average
statistics - frequency
How many times a value occurs in a data set
set
a collection of numbers or other things
elements
The items in the set
|S|
The number of elements in a finite set S
S is a subset of T
all the elements in a set S are also in a set T. This is written as S ⊆ T or by T ⊇ S.
union of two sets A and B
the set of all elements that are each in A or in B or both. The union is written as A ∪ B
intersection of two sets A and B
the set of all elements that are each in both A and B. The intersection is written as A ∩ B
Labels for two sets sharing no elements
disjoint or mutually exclusive
general addition rule for two sets
The number of elements in the union of two finite sets S and T is the number of elements in S, plus the number of elements in T, minus the number of elements in the intersection of S and T.
|S ∪ T| = |S| + |T| - |S ∩ T|
S and T are disjoint, then |S ∪T| =
|S| + |T|, since |S ∩ T| = 0.
Counting Methods - multiplication principle
The number of possible choices of one element apiece from sets A_1, A_2, …, A_n is |A_1| * |A_2| * … * |A_n|
Equations for working with factorials
n! = (n − 1)!(n) and (n + 1)! = (n!)(n + 1)
permutation
A set of n objects has n! permutations
Combinations
event (probability)
A set of an experiment’s possible outcomes
P(E)
The probability P(E) of an event E is a number between 0 and 1, inclusive
E is impossible
P(E) = 0
E is certain
P(E) = 1
P(not E)
1 − P(E)
P(E or F)
P(E) + P(F) - P(E and F)
Events E and F are mutually exclusive
- No outcomes are in E ∩ F.
- The event “E and F” is impossible: P(E and F) = 0
- The special addition rule for the probability of two mutually exclusive events is P(E or F) = P(E) + P(F)
Two events E and F are independent
If neither changes the other’s probability, P(E and F) = P(E)P(F)
P(E and F)
= P(E∩F), P(E|F)*P(F)
E is dependent on F if
P(E|F) <> P(E)
Conditional probability
The probability that E occurs if F occurs.
P(E | F) = |E∩F| / |F|
If each outcome is equally likely, P(E) =
(the number of possible outcomes in E) / (the total number of possible outcomes)
sequence
An algebraic function whose domain contains only positive integers.
A function a(n) that is a sequence can be written as a_n
The domain of an infinite sequence
Set of all positive integers
series
the sum of a sequence’s terms.
Infinite series
is the sum of the sequence’s infinitely many terms, a1 + a2 + a3 +` . . .
Partial sum
The sum of the first k terms of sequence a_n. Can be written as a_1 + . . . + a_k.
Strategy for calculating average
Use a fixed value. Add numbers above / below fixed value. Then subtract below from above. Average that number.
