What would be an example with solution of the Poisson distribution in the frequency of repurchases?
To explain the Poisson distribution in the frequency of repurchases, let's consider an example:
Suppose we are analyzing the number of times a customer makes a repurchase within a certain time frame, such as a week. We want to know the probability of a customer making a certain number of repurchases.
Let's say, on average, we observe 2 repurchases per week. We can use the Poisson distribution to model the frequency of repurchases.
The Poisson distribution has a single parameter, λ (lambda), which represents the average rate of events occurring within a specific time interval.
In our example, λ = 2, as we observe an average of 2 repurchases per week.
To find the probability of a customer making a specific number of repurchases (k) in a week using the Poisson distribution, we can use the following formula:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
- P(X = k) is the probability of k repurchases.
- e is the mathematical constant approximately equal to 2.71828.
- λ is the average number of repurchases per week.
- k is the number of repurchases we want to compute the probability for.
- k! denotes k factorial (k! = k * (k-1) * (k-2) * ... * 2 * 1).
Now, let's calculate the probability of a customer making exactly 3 repurchases in a week using the Poisson distribution:
P(X = 3) = (e^(-2) * 2^3) / 3!
Calculating this expression, we get:
P(X = 3) ≈ 0.180
So, the probability of a customer making exactly 3 repurchases in a week, based on our average rate of 2 repurchases per week, is approximately 0.180 or 18%.
You can similarly calculate the probabilities for different values of k using the formula mentioned above, adjusting the value of λ based on the given situation.