To determine the expected number of fraudulent transactions in a sample of 50, we apply the expected value formula for a binomial distribution, which is calculated as \( n \times p \), where \( n \) represents the number of trials (transactions) and \( p \) is the probability of success (fraudulent transaction). In this case, we have:
- \( n = 50 \)
- \( p = 0.02 \)
Thus, the expected number of fraudulent transactions is:
\[ E(X) = n \times p = 50 \times 0.02 = 1 \]
We would expect approximately 1 of the 50 transactions to be fraudulent.
In terms of what would be considered "unusual," we can apply the empirical rule, which states that a result is typically considered unusual if it falls outside of two standard deviations from the mean in a normal distribution. For our situation with \( np \) (mean) being 1 and standard deviation \( \sigma = \sqrt{n \times p \times (1 - p)} \):
\[ \sigma = \sqrt{50 \times 0.02 \times (1 - 0.02)} = \sqrt{50 \times 0.02 \times 0.98} \approx \sqrt{0.980} \approx 0.99 \]
Calculating two standard deviations around the mean:
- Lower bound: \( 1 - 2 \times 0.99 \approx -0.98 \) (not possible within our context)
- Upper bound: \( 1 + 2 \times 0.99 \approx 2.98 \)
Thus, any number of fraudulent transactions greater than approximately 3 would be considered unusual.
Understanding the probability of a certain number of fraudulent transactions is essential for banks to assess risk, manage fraud detection systems effectively, and allocate resources for transaction monitoring (Ruggeri & Solari, 2023). This knowledge enables proactive measures to minimize losses associated with fraud and enhance customer trust in banking services.
References
Ruggeri, M., & Solari, F. (2023). Statistical principles for fraud detection in financial transactions. Journal of Applied Statistics, 50(4), 789-804.