Question

On the basis of the previous record, an average of 4 accidents in one day occur on the superhighway during peak rush timings. Calculate the following if possible. If any part of question (you feel) is not possible to calculate give reason:
a. Evaluate the probability that on any Particular day there will be fewer than 3 accidents on this superhighway during the peak rush timings.
b. Compute the probability that on any Particular day there will be more than 6 accidents on this Super highway during the peak rush timings.
c. Find the Expected number of accidents per day and variance.

Answer 1

Poisson distribution  is used for independent events which occur at a constant rate within a given interval of time. Hence the following calculations are done.

The provided mean is $\lambda= 4$ .

i)$Pr(X<3)=Pr(X=0)+Pr(X=1)+Pr(X=2)$

=0.0183+0.0733+0.1465

= 0.2381

ii)$Pr(X>6)=1−Pr(X\le6)$

$=Pr(X=0)+Pr(X=1)+Pr(X=2)+Pr(X=3)+Pr(X=4)+Pr(X=5)+Pr(X=6)$

=0.0183+0.0733+0.1465+0.1954+0.1954+0.1563+0.1042

= 0.8893

iii) If μ is the average number of successes occurring in a given time interval or region in the Poisson distribution, then the mean and the variance of the Poisson distribution are both equal to μ.

Hence mean = 4, variance = 4