Term
|
Definition
In deterministic processes, the outcome can be predicted exactly in advance
• Eg. Force = mass x acceleration. If we are given
values for mass and acceleration, we exactly know
the value of force |
|
|
Term
|
Definition
• In random processes, the outcome is not
known exactly, but we can still describe the
probability distribution of possible outcomes
• Eg. 10 coin tosses: we don’t know exactly how
many heads we will get, but we can calculate the
probability of getting a certain number of heads |
|
|
Term
|
Definition
• An event is an outcome or a set of outcomes of
a random process
Example: Tossing a coin three times
Event A = getting two heads = {HTH, HHT, THH}
Example: Picking real number X between 1 and 20
Event A = chosen number is over 8.23 = {X ≤ 8.23}
Example: Tossing a fair dice
Event A = result is an even number = {2, 4, 6}
• Notation: P(A) = Probability of event A |
|
|
Term
|
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
|
Definition
The union of two events A and B is the event that
either A or B or both occurs
[image] |
|
|
Term
|
Definition
The intersection of two events A and B is the event
that both A and B occur[image]
|
|
|
Term
|
Definition
Sample space S of a random phenomenon is the set of all possible outcomes
P(S) for all possibilities in sample space =1
If event A is getting two heads, event A is expressed as a set of outcomes with sample space:
A= {HHTT, HTH, HTTH, THHT, THTH, TTHH
ex: 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
|
Definition
Two events are called disjoint if they can not
happen at the same time
• Events A and B are disjoint means that the
intersection of A and B is zero
Disjoint Rule: If A and B are disjoint events
then:
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
|
Definition
• Events A and B are independent if knowing that A
occurs does not affect the probability that B occurs
• Example: tossing two coins
Event A = first coin is a head
Event B = second coin is a head
• Disjoint events cannot be independent!
• If A and B can not occur together (disjoint), then knowing that
A occurs does change probability that B occurs
• Multiplication Rule: If A and B are independent
P(A and B) = P(A) x P(B)
Independent |
|
|
Term
|
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
|
Definition
A random variable is a numerical outcome of
a 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 or
countable number of distinct values
• Discrete random variables can be summarized
by 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, so
we use a density curve instead of a histogram
X 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
|
Definition
• Average of all possible values of a random
variable (often called expected value)
• Notation: don’t want to confuse random
variables with our collected data variables
µ = mean of random variable
_
x = mean of a data variable
Mean is the sum of all discrete values, with
each 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 dice
µ = 2 ⋅ (1/36) + 3⋅ (2/36) + 4 ⋅ (3/36) + ...+12 ⋅ (1/36) = 252/36 = 7
B/C we have a 1/36 chance of getting a sum of 2 or 12, 2/36 of getting 3 or 11, ect. |
|
|
Term
| Spread of Random Variables |
|
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
|
Definition
The Normal distribution has the shape of a “bell
curve” with parameters µ and σ that determine
the center and spread:"
[image] |
|
|
Term
|
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
|
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 |
|
|
