blems A probability density function is a function that helps one determine the probability that a set of outcomes will occur. For example, suppose we randomly select a student from class and ask their cell phone's typical battery life (in hours). Most students may say between 36 and 48 hours, but some phones have batteries that last 48-60, and others only 8-10. The probability that we would observe an outcome (e.g., battery life) between a and b is p(z) di. where p(I) is the probability density function and I is a potential outcome. In our example, the probability that any given phone's battery life is between 6 and 8 hours would be / p(x)dz. Consider now the probability density function given by p(x) = are z/12. where a > 0 is a constant and a is a cell phone's battery life (in hours). (a) Determine a such that p(2)dr = 1. This asserts, in effect, that there's a 100% chance that a cell phone's battery life is between 0 and co hours. (b) What is the most probable 2-hour interval for cell phone battery life? That is, determine t 2 1 that maximizes the probability that a cell phone's battery life is between t - 1 and t + 1 hours. Use a from part (a)