please answer second question by using the idea of first one

please answer second question by using the idea of first one

3. On the average 4 accidents happen each day in a certain town. What is the probability that tomorrow there will be more than 2 accidents in this town? (Assume Poisson law for rate events.)

Solution

We have given that average 4 accidents happen each day in a certain town.

Let X be a random variable that number of accidents happen in a certain city.

Also given that X ~ Poison ( average () = 4)

The p.m.f of X is,

P(X =x) = ( e^- * ^x ) / x!

Now we have to find the probability that tomorrow there will be more than 2 accidents in this town.

P(X > 2) = 1 - P(X <=2)

= 1 - [ P(X=0) + P(X=1) + P(X=2) ]

P(X=0) =(e^-4 * 4⁰) / 0! = 0.0183

P(X=1) = (e^-4 * 4¹) / 1! =  0.0733

P(X=2) = (e^-4 * 4²) / 2! = 0.1465

P(X >2) = 1 - [ 0.0183 + 0.0733 + 0.1465 ] = 1 - 0.2381 = 0.7619

Now in the next part ,

If X is the waiting time till the next accident the distribution of X is Exponential distribution.

X ~Exp ( 1/)

parameter = 1 / = 1/4 = 0.25

X ~ Exp(0.25)

Mean of X is 1 / = 0.25

and variance of X = 1 / ² = 1 / 4² = 0.0625