Question

In a biomedical study with rats, a dose-response investigation is used to determine the effect of the dose of a toxicant on their survival time. The toxicant is one that is frequently discharged into the atmosphere from jet fuel. For a certain dose of the toxicant, the study determines that the survival time, in weeks, has a gamma distribution with α = 5 and β = 10. What is the probability that a rat survives no longer than 60 weeks?

Answer 1

Solution

Let the random variable X be the survival time (time to death). The required probability is

                         

The integral above can be solved through the use of the incomplete gamma function, which becomes the cumulative distribution function for the gamma dis- tribution. This function is written as

which is denoted as F(6; 5) in the table of the incomplete gamma function in Appendix A.23. Note that this allows a quick computation of probabilities for the gamma distribution. Indeed, for this problem, the probability that the rat survives no longer than 60 days is given by

P(X ≤ 60) = F(6; 5) = 0.715.