Respuesta :
Answer:
The probability that a customer will spend more than 15 minutes total in the bank, given that the customer has already waited over 10 minutes is 0.6065.
Step-by-step explanation:
The random variable T is defined as the amount of time a customer spends in a bank.
The random variable T is exponentially distributed.
The probability density function of a an exponential random variable is:
[tex]f(x)=\lambda e^{-\lambda x};\ x>0[/tex]
The average time a customer spends in a bank is β = 10 minutes.
Then the parameter of the distribution is:
[tex]\lambda=\frac{1}{\beta}=\frac{1}{10}=0.10[/tex]
An exponential distribution has a memory-less property, i.e the future probabilities are not affected by any past data.
That is, P (X > s + x | X > s) = P (X > x)
So the probability that a customer will spend more than 15 minutes total in the bank, given that the customer has already waited over 10 minutes is:
P (X > 15 | X > 10) = P (X > 5)
[tex]\int\limits^{\infty}_{5} {f(x)} \, dx =\int\limits^{\infty}_{5} {\lambda e^{-\lambda x}} \, dx\\=\int\limits^{\infty}_{5} {0.10 e^{-0.10 x}} \, dx\\=0.10\int\limits^{\infty}_{5} {e^{-0.10 x}} \, dx\\=0.10|\frac{e^{-0.10 x}}{-0.10}|^{\infty}_{5}\\=[-e^{-0.10 \times \infty}+e^{-0.10 \times 5}]\\=0.6065[/tex]
Thus, the probability that a customer will spend more than 15 minutes total in the bank, given that the customer has already waited over 10 minutes is 0.6065.
