EX 4.24

Suppose the survival time $X$ in weeks of a randomly selected male mouse exposed to 240 rads of gamma radiation has (what else!) a gamma distribution with $\alpha =8$ and $\beta =15$ . (Data in Survival Distributions: Reliability Applications in the Biomedical Services, by A. J. Gross and V. Clark, suggests $\alpha \approx 8.5$ and $\beta \approx 13.3$ .) The expected survival time is $E\left(X\right)=\left(8\right)\left(15\right)=120$ weeks, whereas $V\left(X\right)=\left(8\right){\left(15\right)}^2=1800$ and ${\sigma }_X=\sqrt{1800}=42.43$ weeks. The probability that a mouse survives between 60 and 120 weeks is

$$P\left(60\le X\le 120\right)=P\left(X\le 120\right)-P\left(X\le 60\right)$$ $$=F\left(120/15;8\right)-F\left(60/15;8\right)$$ $$=F\left(8;8\right)-F\left(4;8\right)=.547-.051=.496$$

The probability that a mouse survives at least 30 weeks is

$$P\left(X\ge 30\right)=1-P\left(X<30\right)=1-P\left(X\le 30\right)$$ $$=1-F\left(30/15;8\right)=.999$$