SOLUTION: A vendor sells merchandise through Amazon and eBay. On eBay she sells an average rate of 17 items per day, while on Amazon the daily average is 19. Both sales follow a Poisson dist

Question 1198955: A vendor sells merchandise through Amazon and eBay. On eBay she sells an average rate of 17 items per day, while on Amazon the daily average is 19. Both sales follow a Poisson distribution and are independent of each other. What is the probability that she sells 35 items on a given day?
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
**1. Determine the Total Average Sales:**
* Total average sales per day: 17 items/day (eBay) + 19 items/day (Amazon) = 36 items/day
**2. Define the Random Variables:**
* Let X be the number of items sold on eBay per day.
* Let Y be the number of items sold on Amazon per day.
* Let Z be the total number of items sold per day (Z = X + Y)
**3. Determine the Distribution of Z**
* Since X and Y are independent Poisson random variables, their sum (Z) also follows a Poisson distribution.
* The mean of Z is the sum of the means of X and Y:
* Mean of Z (λ_Z) = λ_X + λ_Y = 17 + 19 = 36
**4. Calculate the Probability of Selling 35 Items**
* We want to find P(Z = 35), where Z follows a Poisson distribution with mean λ_Z = 36.
* The probability mass function of a Poisson distribution is:
* P(Z = k) = (λ_Z^k * e^(-λ_Z)) / k!
* where:
* k is the number of occurrences (in this case, 35)
* λ_Z is the mean of the Poisson distribution (36)
* e is the base of the natural logarithm (approximately 2.71828)
* k! is the factorial of k
* P(Z = 35) = (36^35 * e^(-36)) / 35!
* **Use a calculator or statistical software to calculate this value.**
**Therefore, the probability that the vendor sells 35 items on a given day is given by the Poisson probability mass function with λ_Z = 36 and k = 35.**
**Note:**
* Calculating this probability directly might involve very large numbers and can be computationally challenging.
* You can use statistical software (like R, Python with libraries like SciPy) or online calculators to efficiently compute the Poisson probability.

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