The dispatcher at a central fire station has observed that the timebetween calls

Question

The dispatcher at a central fire station has observed that the timebetween calls is an exponential random variable random variablewith a mean of 32 minutes.

a) A call has just arrived. What is the probability that thenext call will arrive within the next half hour?

b) What is the probability that there will be exactly two callsduring the next hour?

Explanation / Answer

The dispatcher ata central fire station has observed that the time between calls isan exponential random variable random variable with a mean of 32minutes.     ---->    f(x) = (1/32)*Exp[-x/32]      ---->    F(x) = 1 -Exp[-x/32] a) A call hasjust arrived. What is the probability that the next call willarrive within the next halfhour?     Prob{Next Call< 30 min} = F(30) = 1 - Exp[-(30)/32] = 0.6084 b) What is theprobability that there will be exactly two calls during the nexthour?      Y ~ Poisson{ = (60)/(32) =1.875}      ---->    Prob{Y =k} = (^k)*Exp[-]/(k!)      Prob{Exactly 2 CallsNext Hour} =           = Prob{Y = 2}           = ((1.875)^(2))*Exp[-(1.875)]/(2!)           = 0.2696 .

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