Answer to Question #318048 in Statistics and Probability for Jack

Question

A. Answer the following. Solution is required.

A call center agent in a BPO company has received an average of 15 calls per hour.

a. What is the probability that he will receive 7 calls from 3am to 3:30am? Determine also its mean and variance. (15 points)

b. What is the probability that he will receive atleast 6 calls for a period of 1 hour? (10 points

c. What is the probability that he will receive atleast 8 calls, but not more than 12 calls in a span of 1 and a half hour? (10 points)

B. Give atleast three (3) real-life applications (each) of Binomial distribution and Poisson distribution. (15 points)

Answer 1

Use formula of Poisson

$P=\frac{\lambda^m}{m!}e^{-\lambda}$

$a. \lambda/2=15/2=7.5$

m=7

$P(7)=\frac{7.5^7}{7!}e^{-7.5}=0.146$

$\mu=D=\lambda=7.5$

$\sigma=\sqrt{D}=2.74$

b. $P(X\ge6)=1-P(0)-P(1)-P(2)-P(3)-P(4)-P(5)$

$P(0)=1e^{-15}=0.0000003$

$P(1)=15e^{-15}=0.0000045$

$P(2)=\frac{15^2}{2}e^{-15}=$ 0.0000338

$P(3)=\frac{15^3}{6}e^{-15}=0.0001688$

$P(4)=\frac{15^4}{24}e^{-15}=0.0006328$

$P(5)=\frac{15^5}{120}e^{-15}=0.001894$

$P(X\ge6)=1-0.0027=0.9973$

c.

P(8<X<12,1.5)=P(8,1.5)+P(9,1.5)+P(10,1.5)+P(11,1.5)+P(12,1.5), P(8<X<12,1.5)=P(8,1.5)+P(9,1.5)+P(10,1.5)+P(11,1.5)+P(12,1.5)

$1.5 \lambda=22.5$

$P(8,1.5)=\frac{22.5^8}{8!}e^{-22.5}=0.00028$

$P(9,1.5)=\frac{22.5^9}{9!}e^{-22.5}=0.00069$

$P(10,1.5)=\frac{22.5^{10}}{10!}e^{-22.5}=0.00155$

$P(11,1.5)=\frac{22.5^{11}}{11!}e^{-22.5}=0.00317$

$P(12,1.5)=\frac{22.5^{12}}{12!}e^{-22.5}=0.00595$

$P(8<X<12)=0.00028+0.00069+0.00155+0.00317+0.00595=0.012$

B.

Poison distribution

1. Calls per Hour at a Call Center

Call centers use the Poisson distribution to model the number of expected calls per hour that they’ll receive so they know how many call center reps to keep on staff.

2. Number of Arrivals at a Restaurant

Restaurants use the Poisson distribution to model the number of expected customers that will arrive at the restaurant per day

3. Number of Website Visitors per Hour

Website hosting companies use the Poisson distribution to model the number of expected visitors per hour that websites will receive.

Binomial distribution

1.Number of Side Effects from Medications

Medical professionals use the binomial distribution to model the probability that a certain number of patients will experience side effects as a result of taking new medications.

2. Number of Fraudulent Transactions

Banks use the binomial distribution to model the probability that a certain number of credit card transactions are fraudulent.

3.Number of Spam Emails per Day

Email companies use the binomial distribution to model the probability that a certain number of spam emails land in an inbox per day.