4. The probability that a telephone call lasts no more than t minutes is often m

Question

4. The probability that a telephone call lasts no more than t minutes is often modeled as an exponential cumulative distribution function F(t)={(1-e^(-t/3), t?0,@0 otherwise.)? a) What is the probability density function of the duration in minutes of a telephone conversation? b) What is the probability that a conversation will last between 2 and 4 minutes? c) What is E(T), the expected duration of a telephone call? d) What are the variance and standard deviation of T? e) What is the probability that a call duration is within

Explanation / Answer

The probability that a telephone call lasts no more than t minutes is often modeled as an exponential cumulative distribution function F(t)={(1-e^(-t/3), t> 0, =0 otherwise.) a) f(t) = F'(t) = (e^(-t/3))/3 b) probability that a conversation will last between 2 and 4 minutes = F(4) – F(2) = e^(-2/3) - e^(-4/3) = 0.25 c) lambda = 1/3 , E(T) = 1 / lambda = 3 minutes d) Var(T) = 1/lambda^2 = 9 Standard deviation = 3 minutes e) probability that a call duration is within ±1 standard deviation of the expected call duration = probability call duration is in between 3 + 3 = 6minutes and 3-3 = 0 minutes = F(6) – F(0) = 0.865

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