# Shared Flashcard Set

## Details

Statistics 111; Lectures 8-10- Probability
Terms pertaining to lectures before third homework; Shane T. Jensen, STAT-111 Fall 2010; Introduction to the Practice of Statistics Ch.4 (4.1, 4.2, 4.3) and Ch. 1.3
21
Mathematics
10/25/2010

Term
 Deterministic Processes
Definition
 In deterministic processes, the outcome can be predicted exactly in advance•  Eg. Force = mass x acceleration. If we are givenvalues for mass and acceleration, we exactly knowthe value of force
Term
 Random Processes
Definition
 •  In random processes, the outcome is notknown exactly, but we can still describe theprobability distribution of possible outcomes•  Eg. 10 coin tosses: we don’t know exactly howmany heads we will get, but we can calculate theprobability of getting a certain number of heads
Term
 Event
Definition
 •  An event is an outcome or a set of outcomes ofa random processExample: Tossing a coin three timesEvent A = getting two heads = {HTH, HHT, THH}Example: Picking real number X between 1 and 20Event A = chosen number is over 8.23 = {X ≤ 8.23}Example: Tossing a fair diceEvent A = result is an even number = {2, 4, 6}•  Notation: P(A) = Probability of event A
Term
 Compliment Rule
Definition
 •  The complement Ac of an event A is the event that A does not occur •  Complement Rule : P(Ac) = 1 - P(A) Use compliment rule generally when there are fewer outcomes to calculate for the compliment than for the event. (Getting at least one of something. Easier to calculate compliment of event, getting zero of something, rather than calculate getting 1, 2, 3,...100 different possibilities positively). Look for *at least*   [image]
Term
 Union of Events
Definition
 The union of two events A and B is the event that either A or B or both occurs   [image]
Term
 Intersection of Events
Definition
 The intersection of two events A and B is the event that both A and B occur[image]
Term
 Sample space
Definition
 Sample space S of a random phenomenon is the set of all possible outcomesP(S) for all possibilities in sample space =1If event A is getting two heads, event A is expressed as a set of outcomes with sample space:A= {HHTT, HTH, HTTH, THHT, THTH, TTHHex: S={Heads, Tails} (or S={H, T})S={1, 2, 3, 4}S={ All numbers between 0 and 1}S= {HHTT, HTH, HTTH, THHT, THTH, TTHH}
Term
 Disjoint Events
Definition
 Two events are called disjoint if they can nothappen at the same time•  Events A and B are disjoint means that theintersection of A and B is zeroDisjoint Rule: If A and B are disjoint eventsthen: P(A or B) = P(A) + P(B)Ex. Probability of an accident happening on a weekend (Saturday or sunday) Because an accident can occur on either Saturday or on Sunday but the same accident cannot occur on both days, the events are disjoint. P(Saturday or Sunday)=P(Saturday)+P(Sunday)
Term
 Independent events
Definition
 •  Events A and B are independent if knowing that Aoccurs does not affect the probability that B occurs•  Example: tossing two coinsEvent A = first coin is a headEvent B = second coin is a head•  Disjoint events cannot be independent!•  If A and B can not occur together (disjoint), then knowing thatA occurs does change probability that B occurs•  Multiplication Rule: If A and B are independentP(A and B) = P(A) x P(B)Independent
Term
 Conditional Probability
Definition
 Let A and B be two events •  The conditional probability that event B occurs given that event A has occurred is: P(A and B) P(B | A) = P(A) •  Eg. probability of disease given test positive[image]
Term
 Random Variable
Definition
 A random variable is a numerical outcome ofa random process or random event•  Example: three tosses of a coin•  S = {HHH,THH,HTH,HHT,HTT,THT,TTH,TTT}•  Random variable X = number of observed tails•  Possible values for X = {0,1, 2, 3}
Term
 Discrete Random Variables
Definition
 A discrete random variable has a finite orcountable number of distinct values•  Discrete random variables can be summarizedby their probability distribution•  Random variable X = the sum of two dice•  X takes on values from 2 to 12
Term
 Continuous Random Variable
Definition
 •  Continuous random variables have a non-countable number of values•  Can’t list the entire probability distribution, sowe use a density curve instead of a histogramX takes on all values in an interval of numbers. Assigns probabilities to intervals rather than to individual outcomes. All continuous probability distributions assign probability 0 to every individual outcome (because P is described by area and if x doesn't have a dimension, there cannot be an area)
Term
 Mean of Random Variables
Definition
 •  Average of all possible values of a randomvariable (often called expected value)•  Notation: don’t want to confuse randomvariables with our collected data variablesµ = mean of random variable_x = mean of a data variableMean is the sum of all discrete values, witheach value weighted by its probability:µ = ∑ X i ⋅ P(X i ) = X1 ⋅ P(X1 ) + X 2 ⋅ P(X 2 ) + + X n ⋅ P(X n )i•  Example: X = sum of two diceµ = 2 ⋅ (1/36) + 3⋅ (2/36) + 4 ⋅ (3/36) + ...+12 ⋅ (1/36) = 252/36 = 7B/C we have a 1/36 chance of getting a sum of 2 or 12, 2/36 of getting 3 or 11, ect.
Term
Definition
 •  Spread of all possible values of a random variable around its mean µ •  Again, we don’t want to confuse random variables with our collected data variables: σ = standard deviation of random variable s = standard deviation of a data variable •  SD is based on the sum of the squared deviations away from the mean of all possible values, weighted by the values probability: [image] For rolling dice example: [image]
Term
 Mean/SD of Transformed Random Variables
Definition
 Putting variables into a linear equation to fit via transformation. Often involves a simple addition or subtraction. These do not affect the SD and mean calculations of the random variable
Term
 Combining Random Variables
Definition
 For transformed variable Y = a + b·X mean(Y) = a + b·mean(X) SD(Y) = |b|·SD(X)   •  We can also calculate center and spread of the sum of more than one variable: Z = a + b·X + c·Y   •  The mean formula is easy: mean(Z) = a + b·mean(X) + c·mean(Y)[image]
Term
 Normal Distribution
Definition
 The Normal distribution has the shape of a “bell curve” with parameters µ and σ that determine the center and spread:"   [image]
Term
 Standard Normal
Definition
 Each different value of µ and σ gives a different Normal distribution, denoted N(µ,σ) If µ = 0 and σ = 1, we have the "Standard Normal distribution" [image]
Term
 68-95-99.7 Rule
Definition
 With all normal distributions: •  68% of observations are between µ - σ and µ + σ •  95% of observations are between µ - 2σ and µ + 2σ  •  99.7% of observations are between µ - 3σ and µ + 3σ [image]
Term
 Standardization/ Reverse Standardization
Definition
 Non-standard normal distributions must be transformed into being standard normal so we can use the table. (set µ to 0 and σ to 1) To do this we use: Z= (X−µ)/σ This helps us find a percentage    Reverse Standardization helps us go from a percentage to an X value (e.g. At what length of pregnancy do we find 10% of the population?) We just flip the formula around to find: X =σ ⋅ Z+µ
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