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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 |
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Definition
*the proportion of times the outcome would occur in a very long series of repetitions
*each possible event in the sample space S |
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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 |
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| Sampling with Replacement |
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Definition
| trials are independent only when you put the coin back each time |
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Definition
| a subset of the sample space |
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Definition
| a set, or list, of all possible outcomes of a random process |
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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) |
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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 |
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Term
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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) |
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Definition
*deal with discrete data— data that can only take on a limited number of values
*these values are often integers or whole numbers |
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Term
| We can assign probabilities either: |
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Definition
*Empirically
*Theoretically |
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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 |
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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 |
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Term
| Multiplication Rule for Independent Events |
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Definition
| If A and B are independent, P(A and B) = P(A)P(B) |
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| General Addition Rule for any two events A and B |
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Definition
| P(A or B) = P(A) + P(B) – P(A and B) |
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| Conditional Probabilities |
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Definition
reflect how the probability of an event can change if we know that some other event has occurred/is occurring
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Definition
*important application of conditional probabilities
*foundation of many modern nstatistical applications beyond the scope of this course
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Term
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Definition
a variable whose value is a numerical outcome
of a random phenomenon |
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Term
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Definition
| X has a finite number of possible values |
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| Continuous Random Variable |
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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) |
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Term
| The 68-95-99.7% Rule for Normal Distributions |
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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 |
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Definition
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Term
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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 |
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Term
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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 |
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Term
| Variance of a Discrete Random Variable |
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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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