The gamma distribution is a continuous probability distribut

The gamma distribution is a continuous probability distribution, characterized by shape parameter ? and rate parameter ?. This distribution is often used to model lifetimes of electric components. Let Y follow a gamma distribution with pdf f(y) = ? ?y ??1 e ?y/? ?(?),

where y ? 0, ? > 0 and ? > 0. Notationally, Y ? gamma(?, ?), Suppose the weight of a ream of paper has gamma(2, 0.35) distribution.

(a) Approximately what proportion of the reams will weigh less than 6.5 pounds?

(b) What is the probability that a randomly chosen ream will weigh more than 2.3 pounds?

(c) What is the approximate probability that a randomly chosen ream will weigh between 2.3 and 6.5 pounds?

Solution

Gamma to normal distribution approximation

mean = alpha/beta = 2/0.35 = 5.7143

Variance = alpha/beta^2 = 2/(0.35)^2 = 16.32

s.d = sqrt(variance) = sqrt(16.32) = 4.0406

a.) P(X<6.5) = P(Z<(6.5-5.714)/4.0406) = P(Z<0.194) = 0.5753

b.) P(X>2.3) = P(Z>(2.3-5.714)/4.0406) = P(Z>-0.84) = P(Z<0.84) = 0.7995

c.) P(2.3<X<6.5) = P(-0.84<Z<0.19) = 0.5753 - (1-0.7995) = 0.3748