General Statistics

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1) The expected number of accidents in a certain intersection in one day is 1.5.

a) What is a reasonable probability distribution for the number of accidents that occur in a day?
A Poisson distribution with  = 1.5 accident/day.

b) What's the probability that no accidents occur in a given day?
Use Poisson pdf, Prob(no accident) = p(r=0|,T=1)

c) What is the probability that 2 or 3 accidents occur in a given day?
Prob (r = 2 or r = 3 ) = p(r=2|,T) + p(r=3|,T)

d) What is the probability of 1.5 accidents on a given day?
Zero. The number of accidents has to be an integer.

e) What is the probability no accidents occur over 3 certain consecutive days?
Prob(r=0 over 3 days) = Prob (no accidents on day 1 and no accidents on day 2 and no accidents on day 3)= Prob(no accidents on day 1) x Prob(no accidents on day 2) x Prob(no accidents on day 3)= p(r=0|,T=1)3

f) Installing a traffic light at the intersection would decrease the expected number of accidents per day to 0.75. Associated with each accident is a 5% chance that someone will sue and win $1,000,000. What is the expected savings from the traffic light for a day?
Let Z be number of lawsuits lost on a given day.
When 1.5 accidents are expected per day then 1.5x0.05 = 0.075 lawsuits are expected to be lost per day. When 0.75 accidents are expected per day, 0.75x0.05 = 0.0375, lawsuits are expected to be lost per day. As a result, when the rate is 1.5, the expected loss per day is 0.075x$1,000,000 = $75,000. When the rate is 0.75, the expected loss per day is 0.0375x$1,000,000 = $37,500. The expected savings is $75,000 - $37,500 = $37,500.

2) Earthquakes tend to occur in an area at a rate of one per 24 months. Multiple occurrences of earthquakes in an area are independent.
a) What is a good distribution to model the number earthquakes that occur over a two-year time span ?
A Poisson distribution with  = 1 earthquake per 2 year..

b) What is the probability that 2 or 3 earthquakes in a time span of 2 years
Prob (r = 2 or r = 3 ) = p(r=2|,T) + p(r=3|,T)

c) How many earthquakes would you expect to see in a time span of 3 years.
1.5 (= rate per 2 years * 1.5 = rate per 3 years.)

d) Suppose there was ...