Question

Suppose that the average number of accidents at an intersection is 2 per month.

a) Use Markov’s inequality to find a bound for the probability that at least 5 accidents will occur next month.

b) Using Poisson random variable (λ = 2) calculate the probability that at least 5 accidents will occur next month. Compare it with the value obtained in a).

c) Let the variance of the number of accidents be 2 per month. Use Chebyshev’s inequality to find a bound on probability that at least 5 accidents will occur next month.

Answer 1

a) The probability that atleast 5 accidents will occur next month using Markov's inequality is:

P(X>=5) <= 2/5

b) Using the Poisson distribution the same probability that atleast 5 accidents will occur next month is :

P(X>=5) = 1- P(X=0) - P(X=1) - P(X=2) - P(X=3) - P(X=4)

= 0.0527

In the first part the upper bound if the above probability is 0.1 which is almost twice the poisson probability.

c) Variance = 2 per month,

Thus, Using Chebyshev's inequality we have:

So we can write this as:

P(X>=5) = P(X-2>=5-2)

=P(X-2 >= 3)

we can equate 3 and k*sigma and we get the value of k as:

3 = k*2 or k = 3/2

From the value of k we get the desired probability as:

P(X>=5) <= 1/k^2

P(X>=5) <= 1/(1.5)^2 = 0.4444