Question

There are two newborns, Gary and Eric. The future lifetime of Gary is uniformly distributed between 0 to 60 years. The future lifetime of Eric is uniformly distributed between 0 to 40 years. Their future lifetimes are independent. Calculate the probability that Gary dies first.

Answer 1

Answer: 0.25

Step-by-step explanation:

Given : The  future lifetime of Gary is uniformly distributed with interval [0 years , 60 years].

Then the probability density function for Gary's future lifetime will be:-

$f(x)=\dfrac{1}{60-0}=\dfrac{1}{60}$

The future lifetime of Eric is uniformly distributed with interval [0 years , 40 years].

Then the probability density function for Erin's future lifetime will be:-

$f(x)=\dfrac{1}{40-0}=\dfrac{1}{40}$

Now, the joint density function for Gary and Eric's future lifetime :-

$f(x,y)=f(x)f(y)=\dfrac{1}{40\times60}=\dfrac{1}{2400}$  [∵Their future lifetimes are independent. ]

Now, the probability that Gary dies first is given by :-

$\int^{60}_{0}\int^{40}_{x}f(x,y)\ dy\ dx\\\\=\int^{60}_{0}\int^{40}_{x}\dfrac{1}{2400}\ dy\ dx\\\\=\int^{60}_{0}\dfrac{40-x}{2400}\ dx\\\\=\dfrac{1}{2400}[40x-\dfrac{x^2}{2}]^{60}_{0}\\\\=\dfrac{1}{2400}(2400-\dfrac{3600}{2})=0.25$

Hence, the probability that Gary dies first =0.25