Exam 2 Notes
Chapter 4
Independence
Addition (Joint)
BD Standard Deviation
√
𝜎 = 𝜎 2 = √𝑉𝑎𝑟(𝑥) = √𝑛𝜋(1 − 𝜋)
P(A|B) = P(A)
P(A or B) = P(A) + P(B)
(𝑒 −𝜆 )(𝜆𝑥 )
Multiplication Rule for Independent Events
P(A and B) = P(A) x P(B|A)
Multiplication Rule/Conditional Probability
P(A|B) = P(A and B) / P(B)
Into P(A and B) = P(A|B)P(B)
Mutually Exclusive (Disjoint)
P(AUB) = P(A) + P(B) – P(A∩B)
CanNOT occur at the same time
Collectively Exhaustive
One of the events MUST occur
Ven Diagram
AUB Union of both circles / A∩B
Intersection of circles / (AUB)’ All outside
circles
(A∩B)’ All except intersection
Left Skewed /
Poisson Distribution
𝑃(𝑋 = 𝑥|𝜆) =
𝑥!
x = # of events
𝜆 = Expected # events per unit
(& average # occurrences in interval)
PD = 𝜆 SD = √𝜆
2nd Vars poisson(pdf for certain and cdf
for cumulative)
Chapter 6
Normal Distribution Theoretical Properties
 Symmetrical
mean = median
 Interquartile range is 1.33 SD
 Range is 6 times SD
Normal Probability Density Function
2nd Vars normal(pdf or cdf)
𝑍=
Chapter 5
Right Skewed \
Expected Value/Population Mean
𝜇 = 𝐸(𝑥) = ∑(𝑥𝑖 )𝑃(𝑥 = 𝑥𝑖 )
sum(L1L2)
Discrete Variance
𝜎 2 = ∑[𝑥𝑖 − 𝐸(𝑥)]2 𝑃(𝑥 = 𝑥𝑖 )
Discrete Standard Deviation
𝜎 = √𝜎 2 = √∑[𝑥𝑖 − 𝐸(𝑥)]2 𝑃(𝑥 = 𝑥𝑖 )
2
√𝑠𝑢𝑚((𝐿1 𝑜𝑟 2 – 𝐸(𝑥)) (𝐿1 𝑜𝑟 2 )
Combinations
nCx =
𝑛!
𝑥!(𝑛−𝑥)!
Binomial Distribution
𝑛!
P(X=x|n,π) = (
) (𝜋 𝑥 )(1 − 𝜋)𝑛−𝑥
𝑥!(𝑛−𝑥)!
n = total # events
x = # of events of interest
𝜋 = Prob. of an event of interest
2nd vars binom(pdf or cdf)
BD Mean
𝜇 = 𝐸(𝑥) = 𝑛𝜋

Z-Transformation
(x – mean) / SD
𝑥−𝜇
𝜎
Normal Probability Plot
 X-Values = X
 Y-Values =Z-Values
 Left Skewed
 Right Skewed
Rank/Proportion/Z-Value
Proportion = #/total n
Closest to the probability on the z-chart
Uniform Probability Density
1
𝑓(𝑥) =
𝑖𝑓 𝑎 ≤ 𝑥 ≤ 𝑏 𝑎𝑛𝑑 0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
𝑏−𝑎
a = min
b = max
𝑎+𝑏
UD Mean
𝜇=
(min + max) / 2
UD Variance
2
𝜎 =
2
(𝑏−𝑎)2
12
(max -min)2 / 12
(𝑏−𝑎)2
UD Standard Deviation 𝜎 = √
12