A doubt in finding the expected value of lifetime

Question

Lifetime of a bulb has uniform probability distribution on (2,12). Bulb is replaced upon failure or upon reaching age 10, whichever occurs first.Find the expected value and standard deviation of age of bulb at time of replacement.. I have approached the problem with considering X=age of bulb at time of replacement with interval(2,10) and p.d.f, f(x)=1/8. On finding the expectation the answer obtained is 6. But the answer provided is 6.8 for expected value. Please guide me through the error in my approach.

Answer 1

HINT

Your distribution should have a discrete weight of $20\%$ on the point $10$ years and the rest is uniform, i.e. $$ f(x) = \begin{cases} 0.1, & x \in (2,10) \\ 0.2, & x = 10. \end{cases} $$ Note this way, $$ \int_2^{10} f(x) dx = 0.2 + \int_2^{10} 0.1 dx = 0.2 + 0.8 = 1, $$ where first integral is in the Riemann-Stieltjes sense (allowing to integrate over functions with discrete weights) and the second is in the ordinary Riemann sense.

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Therefore you need a Riemann-Stieltjes integral to compute the expected value, which would be $$ \int_2^{10} x f(x)dx $$ or in terms of regular Riemann integral $$ \mathbb{E}[X] = 0.2\cdot 10 + 0.1\int_2^{10}xdx, $$ can you now complete it?