QUESTION 2 (3 POINTS) A designer of voice mail systems ust design into the syste
ID: 2291550 • Letter: Q
Question
QUESTION 2 (3 POINTS) A designer of voice mail systems ust design into the system a limit of the duration of messages that it will record. The model on which the design is to be based is that the duration of voice messages in minutes] that people like to leave is generally governed by an exponential distribution T(T) (1/to)eou(T), -T/to with parameter to 15 [corresponding to an average length of message of 15 seconds]. Usingthis model, what limit should be placed on messages if 99% of messages that people wish to leave will not be cutoff by an imposed time limit? Repeat for 98% and 99.9% fractions.Explanation / Answer
average, t = 15 seconds. hence, lamda/ rate = 1/15
f(T) = (1/t0) e^ (-T/t0) u (T)
This follows the exponential distribution where P (T >=t) =lam e^ (-lam * t)
P (T < t) = 1- lam e^ (-lam * t)
we want to find t for P (T < t) = 0.99
0.99 = 1 - 1/15 e ^(-1/15*t)
0.15 = e^ (-t/15)
t = ln(0.15) *(-0.15)
t=28.46 seconds
similarly
0.98 = 1 - 1/15 e ^(-1/15*t)
t = 18.06 seconds
similary
0.999 = 1 - 1/15 e ^(-1/15*t)
t = 63 seconds