SOLUTION: A sales firm receives , on average 3 calls per hour on its toll free number. Let x represent the number of calls received in a given hour. Assuming x has a Poisson distribution.

Question 1186832: A sales firm receives , on average 3 calls per hour on its toll free number. Let x represent the number of calls received in a given hour. Assuming x has a Poisson distribution.
For any given hour find the probability that it will receive the following
1. At most 3 calls
2. At least 3 calls
3. 5 or more calls
4. E(X)
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Here's how to solve this problem using the Poisson distribution:
**1. Understanding the Poisson Distribution**
The Poisson distribution is used to model the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independently of the time since the last event.
**Key Parameter:**
* λ (lambda): The average number of events (calls in this case) per interval (hour). In this problem, λ = 3.
**Formula:**
The probability of *k* events occurring in an interval is given by:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
* e is the base of the natural logarithm (approximately 2.71828)
* k is the number of events (calls)
* k! is the factorial of k
**Calculations:**
**1. At most 3 calls:**
We want to find P(X ≤ 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)
* P(X=0) = (e⁻³ * 3⁰) / 0! ≈ 0.0498
* P(X=1) = (e⁻³ * 3¹) / 1! ≈ 0.1494
* P(X=2) = (e⁻³ * 3²) / 2! ≈ 0.2240
* P(X=3) = (e⁻³ * 3³) / 3! ≈ 0.2240
P(X ≤ 3) ≈ 0.0498 + 0.1494 + 0.2240 + 0.2240 ≈ 0.6472
**2. At least 3 calls:**
We want to find P(X ≥ 3) = 1 - P(X < 3) = 1 - [P(X=0) + P(X=1) + P(X=2)]
P(X ≥ 3) = 1 - (0.0498 + 0.1494 + 0.2240) = 1 - 0.4232 ≈ 0.5768
**3. 5 or more calls:**
We want to find P(X ≥ 5) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)]
* P(X=4) = (e⁻³ * 3⁴) / 4! ≈ 0.1680
P(X ≥ 5) = 1 - (0.0498 + 0.1494 + 0.2240 + 0.2240 + 0.1680) = 1 - 0.8152 ≈ 0.1848
**4. E(X):**
The expected value (mean) of a Poisson distribution is equal to λ.
E(X) = λ = 3
**Answers:**
1. At most 3 calls: ≈ 0.6472
2. At least 3 calls: ≈ 0.5768
3. 5 or more calls: ≈ 0.1848
4. E(X) = 3

RELATED QUESTIONS
a sales firm receives an average of four calls per hour on its toll free number for any... (answered by robertb)

A business firm has a 24-hour toll-free telephone hotline. Calls arrive at the rate of... (answered by Boreal)

A High Speed Internet Service Provider (ISP) receives service calls from its clients on a (answered by ewatrrr)

A toll free number is available from 9AM to 9PM to your customers to register complaints... (answered by mathmate)

During normal business hours on the east coast, calls to the toll-free reservation number (answered by Boreal)

Please help with these questions urgently. Question 4 (7 marks = 1 + 1 + 1 + 1 + 1 + 1 (answered by psbhowmick)

Sports Score Hot Line Calls (Optional) Sports Scores Hot Line receives, on the average, 8 (answered by mathmate)

(1). Jack and Jill work in the school office. They are recording and analysing the number (answered by mathmate)

Recent research suggests that Americans make an average of 10 phone calls per day. Let... (answered by Boreal)