The annual rainfall in Cleveland, Ohio, is normally distributed with mean 40 inc

Question

The annual rainfall in Cleveland, Ohio, is normally distributed with mean 40 inches and standard deviation 8 inches. Let Xi denote the rainfall in year i (i = 1, 2, 3, 4) and assume that rainfall is independent from year to year.

(a) Find the probability that the sum of the rainfall in the next four years exceeds 164 inches.

(b) Use Chebyshev’s inequality to estimate P(16 < X1 < 64) (the probability that the next year’s rainfall is between 16 and 64 inches). Also calculate this probability exactly using the normal distribution.

Explanation / Answer

The annual rainfall in Cleveland, Ohio, is normally distributed with mean 40 inches and standard deviation 8 inches. Let Xi denote the rainfall in year i (i = 1, 2, 3, 4) and assume that rainfall is independent from year to year.

(a) Find the probability that the sum of the rainfall in the next four years exceeds 164 inches.

Mean for 4 years=40+40+40+40=160

Variance = s12+ s22+ s32+ s42 =64+64+64+64=256

Sd=16

Z vale for 164, z=(164-160)/16 =0.25

P( x >164) = P( z >0.25)

= 0.4013

(b) Use Chebyshev’s inequality to estimate P(16 < X1 < 64) (the probability that the next year’s rainfall is between 16 and 64 inches).

16 is below 3 sd and 64 is above 3 sd

The required probability = 1-1/32 = 0.8889

Also calculate this probability exactly using the normal distribution.

Z value for 16, z=(16-40)/8 =-3

Z value for 64, z=(64-40)/8 =3

P(16 < X1 < 64) =P( -3<z<3)

=P(z <3) –P(z < -3)

= 0.9987 - 0.0013

=0.9974

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