Question

It is raining in Chicago. About 50% of the drops land downtown. Downtown happens to be a perfectly circular space around city center. Assume the coordinates of the raindrops are independent and distributed according to the standard normal (~N(0,1)) about the city center.

What is the percentage of drops that land within a radius twice that of downtown?

Answer 1

We know that Downtown is around the city center and the raindrops are distributed normally around the city center as well. Hence, we need to find the z-values or the radius around the center of the normal distribution whose area is 0.50

Find z0 when P(-z0 < z < z0) = 0.50

2 * P(0 < z < z0) = 0.50

P(0 < z < z0) = 0.25

P(z < z0) = 0.50 + 0.25 = 0.75

From the z-table, z0 = 0.675 which is the radius of downtown

Now, we need to find the percentage of drops that land within a radius twice that of downtown OR P(-2*z0 < z < 2*z0)

P(-1.35 < z < 1.35) = 2*P(0 < z < 1.35) = 2*( P(z < 1.35) - 0.50 ) = 2*(0.9115 - 0.50) = 2*0.4115 = 0.8230

82.30% of the drops land within a radius twice that of downtown