Key Terms
Discrete
You COUNT. Number of books in a bag.
Continuous
You MEASURE. Weight of a book.
Symbol
P(X <= x) The cdf gives area to the LEFT of a value. It answers: "What is the probability that X is less than or equal t
Area to the right
P(X > x) = 1 - P(X <= x) Area between two values c and d: P(c < X < d) = P(X < d) - P(X < c)
Notation
X ~ Exp(m) m = decay parameter (NOT the mean -- this is a frequent error)
Example
A machine has been running without failure for 10 years. The probability it fails in the next 7 years is the same as it
Mean
Mu = (0 + 15) / 2 = 7.5 minutes Std deviation: sigma = sqrt(225/12) = 4.3 minutes 90th percentile: k = 0.90 * 15 = 13.5
Common applications
Time between customer arrivals, product lifetimes, time between natural events.
Given
Average wait time = 4 minutes. Then m = 1/4 = 0.25.
Shortcut formula
K = ln(area to left) / (-m)
Formal statement
P(X > r + t | X > r) = P(X > t) for all r >= 0, t >= 0
Every probability and percentile problem
Sketch the distribution, shade the region, then calculate. This is explicitly required in the source text and prevents e
UNIFORM
X ~ U(a, b)
EXPONENTIAL
X ~ Exp(m)
POISSON
X ~ Poisson(lambda)