Question

2. (20 pts) At a bank with only one teller, the serving time of a customer is a random variable $T$, exponentially distributed with parameter $\lambda$ (i.e. $f_T(t) = \lambda e^{-\lambda t}$, $t \ge 0$), where $\lambda$ depends on the customer and is a continuous random variable uniformly distributed in $[1, 3]$. (You may find the following useful: $\int_{x=a}^{x=b} xe^{-ux} \cdot dx = (-\frac{1}{u}xe^{-ux})|^{x=b}_{x=a} + (-\frac{1}{u^2}e^{-ux})|^{x=b}_{x=a}$) (a) (7 pts) What is the average serving time of a customer? (b) (5 pts) Ali enters the bank and does not know the number of customers in front of him waiting to be served, but assumes that the number of customers in front of him is between 6 and 10 (including 6 and 10) with equal probability. Based on Ali's assumption, what is the average time he has to wait before the teller starts serving him ? (c) (8 pts) If a customer's serving time was 1 (unit of time), is it more probable that this customer's parameter $\lambda$ was greater or smaller than 2? Provide sufficient justification for your answer. (Hint: $(2e^{-1} - 3e^{-2}) > (3e^{-2} - 4e^{-3})$)

          2. (20 pts) At a bank with only one teller, the serving time of a customer is a random variable $T$,
exponentially distributed with parameter $\lambda$ (i.e. $f_T(t) = \lambda e^{-\lambda t}$, $t \ge 0$), where $\lambda$ depends on
the customer and is a continuous random variable uniformly distributed in $[1, 3]$.
(You may find the following useful: $\int_{x=a}^{x=b} xe^{-ux} \cdot dx = (-\frac{1}{u}xe^{-ux})|^{x=b}_{x=a} + (-\frac{1}{u^2}e^{-ux})|^{x=b}_{x=a}$)
(a) (7 pts) What is the average serving time of a customer?
(b) (5 pts) Ali enters the bank and does not know the number of customers in front of him
waiting to be served, but assumes that the number of customers in front of him is between
6 and 10 (including 6 and 10) with equal probability. Based on Ali's assumption, what is
the average time he has to wait before the teller starts serving him ?
(c) (8 pts) If a customer's serving time was 1 (unit of time), is it more probable that this
customer's parameter $\lambda$ was greater or smaller than 2? Provide sufficient justification for
your answer. (Hint: $(2e^{-1} - 3e^{-2}) > (3e^{-2} - 4e^{-3})$)

2. (20 pts) At a bank with only one teller, the serving time of a customer is a random variable T,
exponentially distributed with parameter λ (i.e. fT(t) = λ e^-λ t, t ≥ 0), where λ depends on
the customer and is a continuous random variable uniformly distributed in [1, 3].
(You may find the following useful: ∫x=a^x=b xe^-ux· dx = (-(1)/(u)xe^-ux)|^x=bx=a + (-(1)/(u^2)e^-ux)|^x=bx=a)
(a) (7 pts) What is the average serving time of a customer?
(b) (5 pts) Ali enters the bank and does not know the number of customers in front of him
waiting to be served, but assumes that the number of customers in front of him is between
6 and 10 (including 6 and 10) with equal probability. Based on Ali's assumption, what is
the average time he has to wait before the teller starts serving him ?
(c) (8 pts) If a customer's serving time was 1 (unit of time), is it more probable that this
customer's parameter λ was greater or smaller than 2? Provide sufficient justification for
your answer. (Hint: (2e^-1 - 3e^-2) > (3e^-2 - 4e^-3))

Added by Juan Luis L.

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