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# Mean calculation

There are 3 bulbs in a room. If we switched on all of them. What is the total expected time till the room remains lit? Assume the "on" time for each bulb is an exponential random variable with λ=1 hour

Note by Anurag Choudhary
4 years, 4 months ago

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You want the expected value of the last order statistic $$X_{(3)} = \max\{X_1, X_2, X_3\}$$ of the IID random variables $$X_1, X_2, X_3 \sim {\rm Exponential}(\lambda)$$. The cumulative distribution of $$X_{(3)}$$ is easy to find: it is simply \begin{align*} F_{X_{(3)}}(x) &= \Pr[\max\{X_1, X_2, X_3\} \le x] \\ &= \Pr[X_1 \le x]\Pr[X_2 \le x]\Pr[X_3 \le x] \\ &= (1 - \lambda e^{-\lambda x})^3. \end{align*} With $$\lambda = 1$$, this becomes $F_{X_{(3)}}(x) = (1 - e^{-x})^3,$ so its survival is $S_{X_{(3)}}(x) = 1 - (1 - e^{-x})^3.$ Now recalling that the expected value of a continuous random variable whose support is positive is simply the integral of its survival function, we immediately obtain ${\rm E}[X_{(3)}] = \int_{x=0}^\infty 1 - (1-e^{-x})^3 \, dx = \frac{11}{6}.$ Or you could find the density $$f_{X_{(3)}}(x) = F'_{X_{(3)}}(x)$$ and compute the integral of $$x f(x)$$, but that's just extra work.

- 4 years, 4 months ago