4.44.8
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Univeriable data
data from one variable

Bivariate data
grouped and analyzed data from two variables of a population

Contingency table or twoway table
 a frequency distribution for bivarite data.
 Cells: small boxes inside the rectangleformed by the heavy lines
 row  horizontal
 column  vertical
 Colculating total number: by summing the row totals, or column totals or the frequencies in the 20 cells of cont. table.

Marginal probabilities
 they correspond to events represented in the margin of the contingency table
 Total of the column or a row devided by the total of all columns or rows
 ex. P(A1) = f/N = 381/1164=0.327

Joint probability
 probabilities for joint events
 P(A1&R2) = f/N = 3/1164 = 0.003 in the cells
 Joint probability distribution: joint probabilities are displayed instead of frequencies
 The row and column labels "total" add up to 1.000 and replaced to P(Ai) and P(Ri)

Conditional probability
 P(B/A) "the probability of event of B given A"
 A is the given event
 P(B/A)=P(A&B)
 P(A)

General multiplication rule
P(A&B)= P(A) *P(B/A) writinga tree diagrom is helpful when applying general multiplication rule
 a formula for computing joint probabilities in terms of mariganl and conditional probabilities.
 When the joint and marginal probabilities are known or can be easily determined directly, we use the conditonal probability rule to obtain conditional probabilities.
 When marginal and conditional probabilities are known or can be easily determined directly, we use the general multiplication rule to obtain joint probabilities.


Independet events
 Event B is independent of event A if P(B/A) = P(B)
 If two events are not independent then they are dependent events

Special multiplication rule (for 2 independent events)
 IF A and B are independent events, then
 P(A & B) = P(A)* (P(B)
 IF P(A & B) equal to P(A)* (P(B), then A and B are independent events

Special multiplication rule (for 3 or more indep. events)
 If events A, B, C,.... are independent, then
 P(A&B&C&...)= P(A) * P(B) * P(C)

The Basic Counting Rule (BCR)
 r  different actions
 Suppose that r actions are to be performed in a definite order. Further suppose that there are m1 possibilities for the first action and that corresponding to each of these possibilites are m2 possibilities for the second action, and so on. Then there are m1*m2....mr possibilities altogether for the r actions.
 ex. on pg 214
 number of r=2 selectinga a model and elevation
 there are 4 possibilities for model , m1=4
 3 posibilities for elevation, m2=3
 m1*m2=4*3=12

Factorials
 k  is a positive integer(counting number
 k factorial denoted as k!
 The factorial of a counting number is obrtained by successeively multiplying it by the next smaller counting number until reaching 1.
 k! = k(k1)....2times 1
 0! =1
 ex. 4!=4*3*2*1 = 24

The Permutations Rule
 The number of possible permutations of r objects from a collection of m objects is given by the formula
 mPr= m!
 (mr)!
 the order metter
 AB and BA are different
 The number of possible permutations of r objects that can be formed from a collection of m ojbects is denoted mPr

Special permutation rule
 Finding the number of possible permutations among themselves.
 mPm = m!/(mm)!= m!/0!=m!/1=m!
 the number of possible permutations of m ojbects among themselves is m!

The Combinations rule
 A combination of r objects froma collection of m objects is any unoreded arrangement of r of the m ojbects  any subset of r objects from the collection of m objects, denoted mCr (m choose r)
 AB and BA are the same  list only once
 formula:
 mCr = m!
 r!(mr)!