Four tigers in a reserve forest are monitored using geo tags. The waiting times for responses from 4 tigers in the reserve follow an iid exponential distribution with mean 3. If the system has to locate all 4 tigers within 5 minutes, it has to reduce the expected response time of each geo tag. What is the maximum expected response time that will produce a location for all four tigers within 5 minutes or less with at least $90%$ probability.
I don’t understand this question, can someone explain what it is asking for and provide the solution?
The first step is to find the distribution of the max of exponential RVs. Let $X_n \sim exp(\lambda)$ be i.i.d. RVs and let $Y = \max_n \{X_n\}$.
$P(Y \leq y)$
$=P(\max_n \{X_n\} \leq y)$
$=P(X_1 \leq y, \; X_2 \leq y, \; ..., \; X_n \leq y)$
$=P(X_1 \leq y)^n$ (using independence)
$=(1-e^{-\lambda y})^n$ (using definition of exponential CDF)
In your example, $n=4$, so you can compute $\lambda$ from the inequality $P(Y \leq 5) \geq 0.9$, i.e. $(1-e^{-5 \lambda})^4 \geq 0.9$.