Answer to Question #149680 in Statistics and Probability for Randal Rodriguez

Question

The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the density function below. Find the probability that a bottle of this medicine will have a shell life of

f(x)= {20000/(x+100)^3, x>0,
.........0,(included on the bracket) elsewhere.

(a) at least 365 days. (Decimal form)
(b) at most 365 days. (Decimal form)

Answer 1

a. Let X be the shelf life for a bottle of the medicine, i.e the random variable with the density function f.

$P(x\geq365)$$\space =\int^{\infin}_{x=365}f(x)$

$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=\int^{\infin}_{x=365}\frac{20000}{(x+100)^3}$

$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=(-\frac{10000}{(x+100)^2})\mid^{\infin}_{365}$

$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=0.04624812117$

$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=0.0462$

b.

$P(x\leq365)=1-P(x\geq365)$

$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=1-\int^{\infin}_{x=365}f(x)$

$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=1-\int^{\infin}_{x=365}\frac{20000}{(x+100)^3}$

$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=1-(-\frac{10000}{(x+100)^2})\mid^{\infin}_{365}$

$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=1-\frac{10000}{(365+100)^2}$

$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=1-\frac{400}{8649}$

$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=0.95375187883$

$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space=0.9538$