Probability Distributions Discrete Probability Distribution

– Discrete vs. continuous random variables» discrete - only a countable number of values

» continuous - any value in an interval

– distribution is a table, graph, or formula that gives the probability of observing each value of x.

Binomial Distribution– Conditions (see p.270)

» 1. n units sampled from population with replacement

» 2. Either success or failure

» 3. Probability of success is» 4. Independent samples

» 5. Binomial random variable: x = successes

– distribution

– mean, variance» = n» 2 = n(1-)

xnx

x

nxp

1

Applications– Weekly compliance sampling

» agency wants compliance 95% of time

» 52*0.05 = 2.6

» agency will allow only 2 violations per year, any more and they’ll shut you down

» you run your operation at 95% compliance, what are your odds of being shut down?

Poisson Distribution– Rare event

– characteristics (283)» # of times an even occurs during a unit

» prob same for all units

» # in one unit independent of others

– distribution

– Long-term average of 6 accidents per year at intersection

» probability of exactly 6 this year

» probability of 3 or less

» after improvements, had only 2 accidents. Did the improvements help.

!x

exp

x

False positive example– In a blood test, you test positive for the

presence of the HIV virus. Should you be concerned?

» Test is 99% accurate

» 1 out of 1000 people have HIV

» Out of 100,000

# of false positives =

# of true positives =