5. Binomial Distribution & Discrete Random Variables
Poisson Distribution
Struggling with Statistics for Business?
Join thousands of students who trust us to help them ace their exams!Watch the first videoMultiple Choice
A small electronics retailer tracks the number of customers who arrive to purchase replacement phone chargers. Based on historical data, the store finds that, on average, 3 customers per day buy a charger. The store manager wants to use this information to optimize inventory decisions and reduce the risk of stockouts.
(B) If the store stocks 5 chargers per day, find the probability that they will have inventory remaining on a given day.
A
B
C
D
Verified step by step guidance1
Step 1: Recognize that this problem involves a Poisson distribution. The Poisson distribution is used to model the number of events (e.g., customer arrivals) occurring in a fixed interval of time or space, given a known average rate (λ). Here, the average rate (λ) is 3 customers per day.
Step 2: Define the random variable X as the number of customers who arrive to purchase chargers in a day. X follows a Poisson distribution with parameter λ = 3. The probability mass function (PMF) of a Poisson distribution is given by: P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of events (customers) and e is the base of the natural logarithm.
Step 3: To find the probability that the store will have inventory remaining, calculate the probability that fewer than 5 customers arrive in a day. This is equivalent to finding P(X ≤ 4). Use the cumulative probability formula: P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4).
Step 4: For each value of k (0 through 4), calculate P(X = k) using the PMF formula: P(X = k) = (3^k * e^(-3)) / k!. Compute these probabilities and sum them to find P(X ≤ 4).
Step 5: Once P(X ≤ 4) is calculated, interpret the result. This value represents the probability that the store will have inventory remaining (since fewer than 5 customers arrive). The complement, P(X > 4), would represent the probability of a stockout.
4:00mWatch next
Master Introduction to the Poisson Distribution with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice
