6.5 The pH, a measure of the acidity of water, is important in studies of acid rain. For a certain Florida lake, baseline measurements on acidity are made so any changes caused by acid rain can be noted.

The pH of water samples from a lake is a random variable X with probability density function f (x) =

L 3

8

(7 - x)2 5 … x … 7 0 elsewhere f (x) =

L 3

64 x2 (4 - x) 0 … x … 4 0 elsewhere f (x) =

L 3

32

(x - 2) (6 - x) 2 … x … 6 0 elsewhere

= …

… …

… …

a Sketch the curve of f(x).

b Find the distribution function F(x) for X.

c Find the probability that the pH will be less than 6 for a water sample from this lake.

d Find the probability that the pH of a water sample from this lake will be less than 5.5 given that it is known to be less than 6.