Question

# M. F. Driscoll and N. A. Weiss discussed the modeling and solution of problems concerning motel reservation networks in “An Application of Queuing The

Modeling data distributions
M. F. Driscoll and N. A. Weiss discussed the modeling and solution of problems concerning motel reservation networks in “An Application of Queuing Theory to Reservation Networks” (TIMS, Vol. 22, No. 5, pp. 540–546). They defined a Type 1 call to be a call from a motel’s computer terminal to the national reservation center. For a certain motel, the number, X, of Type 1 calls per hour has a Poisson distribution with parameter $$\displaystyle\lambda={1.7}$$.
Determine the probability that the number of Type 1 calls made from this motel during a period of 1 hour will be:
a) exactly one.
b) at most two.
c) at least two.
(Hint: Use the complementation rule.)
d. Find and interpret the mean of the random variable X.
e. Determine the standard deviation of X.

2021-03-08

Step 1
Given:
$$\displaystyle\lambda={1.7}$$
Formula Poisson probability:
$$P(X=k)=\frac{\lambda^{k}e^{-\lambda}}{k!}$$
Complement rule:
$$\displaystyle{P}{\left(\neg\ {A}\right)}={1}\ -\ {P}{\left({A}\right)}$$
$$\displaystyle{P}{\left({A}\ {\quad\text{or}\quad}\ {B}\right)}={P}{\left({A}\right)}\ +\ {P}{\left({B}\right)}$$
Step 2
SOLUTION
a) Evaluate the formula of Poisson probability at $$\displaystyle{k}={1}:$$
$$\displaystyle{P}{\left({X}={1}\right)}={\frac{{{1.7}^{{{1}}}{\mid}{e}^{{-{1.7}}}}}{{{1}!}}}\approx\ {0.3106}$$
Step 3
b) Evaluate the formula of Poisson probability at $$\displaystyle{k}={0},\ {1},\ {2}:$$
$$\displaystyle{P}{\left({X}={0}\right)}={\frac{{{1.7}^{{{0}}}{e}^{{-{1.7}}}}}{{{0}!}}}\approx\ {0.1827}$$
$$\displaystyle{P}{\left({X}={1}\right)}={\frac{{{1.7}^{{{1}}}{e}^{{-{1.7}}}}}{{{1}!}}}\approx\ {0.3106}$$
$$\displaystyle{P}{\left({X}={2}\right)}={\frac{{{1.7}^{{{2}}}{e}^{{-{1.7}}}}}{{{2}!}}}\approx\ {0.2640}$$
$$\displaystyle{P}{\left({X}\leq{2}\right)}={P}{\left({X}={0}\right)}+{P}{\left({X}={1}\right)}+{P}{\left({X}={2}\right)}$$
$$\displaystyle={0.1827}+{0.3106}+{0.2640}$$
$$\displaystyle={0.7572}$$
Step 4
c) Evaluate the formula of Poisson probability at $$\displaystyle{k}={0},{1},{2}:$$
$$\displaystyle{P}{\left({X}={0}\right)}={\frac{{{1.7}^{{{0}}}{e}^{{-{1.7}}}}}{{{0}!}}}\approx\ {0.1827}$$
$$\displaystyle{P}{\left({X}={1}\right)}={\frac{{{1.7}^{{{1}}}{e}^{{-{1.7}}}}}{{{1}!}}}\approx\ {0.3106}$$
$$\displaystyle{P}{\left({X}{<}{2}\right)}={P}{\left({X}={0}\right)}+{P}{\left({X}={1}\right)}$$
$$\displaystyle={0.1827}+{0.3106}$$
$$\displaystyle={0.4932}$$
Use the complement rule:
$$\displaystyle{P}{\left({X}\geq{2}\right)}={1}-{P}{\left({X}{<}{2}\right)}={1}-{0.4932}={0.5068}$$
Step 5
d) The meanof the Poisson distribition is equal to the value of the parameter $$\displaystyle\lambda$$
$$\displaystyle\mu_{{{X}}}=\lambda={1.7}$$
On average, there are 1.7 Type 1 calls are made from this motel during a period of 1 hour.
Step 6 e) The variance of the Poisson distribution is equal to the value of the parameter $$\displaystyle\lambda$$
$$\displaystyle{\sigma_{{{X}}}^{{{2}}}}=\lambda={1.7}$$
The standard deviation is the square root of the variance:
$$\displaystyle\sigma_{{{X}}}=\sqrt{{{\sigma_{{{X}}}^{{{2}}}}}}=\sqrt{{{1.7}}}\approx\ {1.3038}$$