Answer:
(i) 2/7 or 0.2857 of the time
(ii) 70 days
Step-by-step explanation:
(i) There are two possible conditions, two bulbs working or one bulb working. The long run fraction of time that there is exactly one bulb working is given by the expected time with one light bulb working divided by the expected time with one or two bulbs working:
[tex]F = \frac{E(X=1)}{E(X=1)+E(X=2)}\\ F=\frac{0.05^{-1}}{0.05^{-1}+0.02^{-1}} \\F=\frac{2}{7}=0.2857[/tex]
The long-run fraction of time that there is exactly one bulb working is 2/7 or 0.2857 of the time.
(ii) The expected time between light bulb replacements is the expected time for both bulbs to go out:
[tex]T=E(X=1)+E(X=2) = 0.02^{-1}+0.05^{-1}\\T= 70\ days[/tex]
The expected time between replacements is 70 days.
