Pop SD:
Pop Var(X):
Z-score:
IQR:
Beyond these limits (outliers):
Mutually Exclusive
A and B cannot occur simultaneously.
Right-Skew (+)
Mode < Median < Mean
Left-Skew (-)
Mean < Median < Mode
Sample Var(X):
Independent Events
Events A and B can occur simultaneously, but
they do not affect the probability of one another.
Additive Law (Either A or B, or Both) Intersection
Multiplicative Law (A and B) Union
Complement Rule
"Exactly one"
"At least one success"
Find outcome of 0 successes
and subtract from 1
Bayes' Theorem (We know P(A|B), we need P(B|A)
Ex. Flip coin 5 times
Random Variables
PDF= p(x) = P(X=x) | CDF = P(X≤x)
Binomial RV
1.
2.
3.
4.
Properties of Variance
When X and Y are independent,
Var (X+Y) = Var(X) + Var(Y)
When not independent,
Var (X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)
Fixed # of Trials
Each trial -> only 2
outcomes
P(success) is same in
every trial
Trials are independent
Hypergeometric (w/o replacement)
•
•
•
•
N = population size
n = sample size
r = # of successes in population
x = # of successes in sample
Poisson Distribution
• No upper limit on x
• E(X) = Var(X) = λ
• To get λ, multiply rate per time unit
for the appropriate duration
Properties of Expectation
Continuous Random Variables
P(X < x) is the same as P(X ≤ x)
----> P(a ≤ x ≤ b) = P(x ≤ b) - P(x ≤ a)
For normal distribution (standard), denoted by z:
• µ=0
• σ=1
Random Sampling
E(x̄) = µ
Var(x̄) = σ² / n
The SD of a sample statistic is the standard error
• Smaller standard error means the sample estimate
is likely closer to the true population value
P (X > a)
P (X < a)
P (a < X < b)
"Exceeds," "more
than," "beats target."
Shade the right tail.
"Loses money," "falls
short," "at base." Shade the
left tail. Read it directly.
"Within 1 SD," a "band,"
a "tier." Shade the
middle; subtract 2 areas
Given area -> find x
"97th percentile." You
know shaded area; find z
first, then x = µ + (σ)(z)