Respuesta :
Answer:
The probability that the arrival time between customers will be 12 or less is 0.6027.
Step-by-step explanation:
We are given that the time between arrivals of customers at the drive-up window of a bank follows an exponential probability distribution with a Mean of 13.
Let X = time between arrivals of customers
The probability distribution for exponential distribution is given by;
[tex]f(x) = \lambda e^{-\lambda x} ; x >0[/tex]
where, [tex]\lambda[/tex] = parameter of this distribution or the arrival rate
Since, the mean of exponential distribution = E(X) = [tex]\frac{1}{\lambda}[/tex]
So, 13 = [tex]\frac{1}{\lambda}[/tex] , [tex]\lambda=\frac{1}{13}[/tex]
So, X ~ Exp( [tex]\lambda=\frac{1}{13}[/tex] )
Now, to find the less than or greater than probabilities in exponential distribution we use the Cumulative distribution function of exponential function, i.e.;
[tex]F(x) = P(X \leq x) = 1 - e^{-\lambda x} ; x >0[/tex]
So, probability that the arrival time between customers will be 12 or less is given by = P(X [tex]\leq[/tex] 12)
P(X [tex]\leq[/tex] 12) = [tex]1 - e^{-\lambda x}[/tex]
= [tex]1 - e^{-\frac{1}{13} \times 12}[/tex]
= [tex]1 - e^{-\frac{12}{13} }[/tex] = 0.6027
Therefore, probability that the arrival time between customers will be 12 or less is 0.6027.
