Problems 1. (16 points) Lead contamination of drinking water is a potentially serious public health concern, and can be caused by corrosion of certain types of plumbing fixtures. Lead concentrations in water are often reported in parts per billion (ppb), where 1 ppb is equal to 0.001 mg / L. In the US, the Environmental Protection Agency requires certain actions to be taken if the amount of lead in public water systems exceeds 15 ppb. Suppose that a drinking fountain in a certain library contains lead, but that water usage patterns and environmental factors create some variability in the lead concentration of water samples taken from the fountain. Let X be the lead concentration of the water at the drinking fountain, in ppb. Suppose that X has the following pdf. f(x) = $1030 ifOS x s 10 0 otherwise This density function is chosen because the negative coefcient on x makes it so that higher lead concentrations are relatively less likely, but the numbers and shape of the density function are not meant to be an accurate representation of the amount of lead in any particular drinking fountain. 1.1. A water sample is taken from this drinking fountain, and its lead concentration is measured by a test that cannot detect lead levels below 1 ppb. What is the probability that the test will not detect any lead? 1.2. Three samples are taken independently from the drinking fountain. Let Y be the number of samples containing a detectable amount of lead. What is the distribution of Y? 1.3. What is the probability that exactly two of the three samples will contain a detectable amount of lead? 1.4. What is the probability that at least one of the three samples will contain a detectable amount of lead