A patient is sitting in the waiting room of a doctor's office. We assume that its waiting time in minutes is exponentially distributed with parameter $\lambda = 0.2$. Within what time will the patient be treated with probability $0.9$? The patient waited $5$ minutes without being called. How long does he have to wait with probability $0.9$?
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I have done the following: $$P(X\leq x)=1-e^{-\lambda x}\Rightarrow 0.9=1-e^{-0.2x} \Rightarrow e^{-0.2x} =0.1 \Rightarrow x \approx 11.513$$ Within the first $11.5$ minutes the patient will be treated with probability $0.9$.
Is at the second question "The patient waited $5$ minutes without being called. How long does he have to wait with probability $0.9$?" the answer "He has to wait for 11,5-5 minutes."? Or do we have to calculate something else here?
Hint:
An important property of the exponential distribution is that it lacks memory, that is, if $X$ has an exponential distribution, then, $$P(X>s+t|X>t) = P(X>s)$$
Can you take it from here?