Question

There are on average 12.0 small airplane crashes in the US each year.

1.What is the standard deviation of the number of small airplane crashes that will happen in the US in the next 3 years?

2.What is the probability that there will be exactly 10 small airplane crashes in the US in 2021?

3.We will find the probability that there will be fewer than 14 small airplane crashes in the US in 2021, using normal approximation. In the process, we will need to use a table or app to find P(Z ≤ z) for Z standard normal, and letting z be what?

Answer 1

Let X be the random variable that denotes the number of small airplane crashes in the US each year.

Given, there are on average 12 small airplane crashes in the US each year.

Therefore, X follows the poisson distribution with parameter 12.

X Poisson ( = 12)

1. Answer :

Let Y be the random variable that denotes the number of small airplane crashes in the US in 3 years.

Therefore, Y = X + X + X = 3X

Y Poisson (m = 12 * 3 = 36)

The standard deviation of Y is

SD =

      =

      = 6

Therefore, the standard deviation of the number of small airplane crashes that will happen in the US in the next 3 years is 6.

2. Answer :

X Poisson ( = 12)

The pmf of X is

P(X = x) = (e-12 * 12x) / x! ; x = 0, 1, 2, .........

              = 0                        ; otherwise

P(X = 10) = (e-12 * 1210) / 10!

                = 0.1048

Therefore, the probability that there will be exactly 10 small airplane crashes in the US in 2021 is 0.1048

3. Answer :

X Poisson ( = 12)

Therefore, E(X) = 12 and V(X) = 12

Using normal approximation, X Normal ( = 12, 2 = 12)

P(X < 14) = P(X < 13.5)   ..............(using continuity correction)

                = P((X - ) / < (13.5 - 12) / )

                = P(Z < 0.43)           ..........( Z = (X - ) / )

                = 0.6664                  ..........(Value from standard normal table)

Therefore, using normal approximation, the probability that there will be fewer than 14 small airplane crashes in the US in 2021 is 0.6664