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Ch. 2
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25
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Graduate
09/16/2013

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Term
Randomness
Definition
A phenomenon is random if individual outcomes are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions
Term
Probability
Definition

*the proportion of times the outcome would occur in a very long series of repetitions

 

*each possible event in the sample space S

Term
Independent (Two Events)
Definition
if the probability that one event occurs  on any given trial of an experiment is not affected or changed by the occurrence of the other event
Term
Sampling with Replacement
Definition
trials are independent only when you put the coin back each time
Term
Event
Definition
a subset of the sample space
Term
Sample Space (S)
Definition
a set, or list, of all possible outcomes of a random process
Term
Rules of Probability
Definition

Rule 1. 0 ≤ P(A) ≤ 1 for any event A

 

Rule 2. P(S) = 1 

 

Rule 3. Addition rule: If A and B are disjoint events, then  P(A or B)=P(A)+P(B)

 

Rule 4. Complement rule: For any event A, P(AC)=1-P(A)

 

Rule 5. Multiplication rule: If A and Bare independent

events, then P(A and B)=P(A)P(B)

Term
Disjoint
Definition

*if they have no outcomes in common and can never happen together

 

*The probability that A or B occurs is  then the sum of their individual probabilities

 

*P(A or B) = P(A U B)= P(A) + P(B) 

This is the addition rule for disjoint events

Term
Complement
Definition

*any event A is the event that A does not occur, written as Ac

 

*complement rule states that the probability of an event not occurring is 1 minus the probability that is does occur

 

*P(not A) = P(Ac) = 1 − P(A)

Term
Finite Sample Spaces
Definition

*deal with discrete data— data that can only take on a limited number of values

 

*these values are often integers or whole numbers

Term
We can assign probabilities either:
Definition

*Empirically

 

*Theoretically

Term
Empirically
Definition

*from our knowledge of numerous similar past events



*Mendel, the founder of the new science of genetics, discovered the probabilities of inheritance of a given trait from experiments on peas without knowing about genes or DNA

Term
Theoretically
Definition
*from our understanding of the phenomenon and symmetries in the problem

*A 6-sided fair die: each side has the same chance of turning up
 
 *Genetic laws of inheritance based on meiosis process
Term
Multiplication Rule for Independent Events
Definition
If A and B are independent, P(A and B) = P(A)P(B)
Term
General Addition Rule for any two events A and B
Definition
P(A or B) = P(A) + P(B) – P(A and B)
Term
Conditional Probabilities
Definition

reflect how the probability of an event can change if we know that some other event has occurred/is occurring

 

Term
Bayes’s Rule
Definition

*important application of conditional probabilities

 

*foundation of many modern nstatistical applications beyond the scope of this course

 

 

Term
Random Variable
Definition
a variable whose value is a numerical outcome
of a random phenomenon
Term
Discrete Random Variable
Definition
X has a finite number of possible values
Term
Continuous Random Variable
Definition

X takes all values in an interval

 

 

Example: There is an infinity of numbers between 0 and 1 (e.g., 0.001, 0.4, 0.0063876)

Term
The 68-95-99.7% Rule for Normal Distributions
Definition

*About 68% of all observations are within 1 standard deviation (s) of the mean (μ)


 *About 95% of all observations are within 2 s of the  mean μ


*Almost all (99.7%) observations are within 3 s of the mean

Term
normal quantile plot
Definition
Term
Law of Large Numbers
Definition

*As the number of randomly drawn observations (n) in a sample increases, the mean of the sample (x bar) gets closer and closer to the population mean μ

 

*It is valid for any population

Term
variance (σ2x)
Definition

*A weighted average of the squared deviations (X−μX)2 of the variable X from its mean μX

 

*Each outcome is weighted by its probability in order to take into account outcomes that are not equally likely

 

*Larger the variance of X, the more scattered the values of X on average

 

*The positive square root of the variance gives the standard deviation σ of X

Term
Variance of a Discrete Random Variable
Definition
variance σ2 of X is found by multiplying each squared
deviation of X by its probability and then adding all the products
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