1. What is a Probability of Error Calculator?

Definition: This calculator computes the probability of error (Pₑ) between two distributions using the Q-function.

Purpose: It's commonly used in signal processing, statistics, and communication theory to evaluate error probabilities in detection systems.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( P_e \) — Probability of error (dimensionless)
• \( \mu_1, \mu_2 \) — Means of two distributions
• \( \sigma \) — Standard deviation
• \( Q \) — Q-function (standard normal tail probability)

Explanation: The difference between means is normalized by the standard deviation, and the Q-function gives the tail probability.

3. Importance of Probability of Error Calculation

Details: This calculation is crucial for designing communication systems, statistical tests, and any scenario requiring error rate estimation.

4. Using the Calculator

Tips: Enter the means of your two distributions and the standard deviation. The standard deviation must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the Q-function?
A: The Q-function is the tail probability of the standard normal distribution, giving P(X > x) for a standard normal random variable.

Q2: When is this calculation used?
A: Commonly used in digital communications, hypothesis testing, and signal detection problems.

Q3: What if my standard deviation is very small?
A: A small σ will make the z-score larger, resulting in a smaller probability of error.

Q4: How accurate is the Q-function approximation?
A: The approximation used is accurate to about 7 decimal places for z ≥ 0.

Q5: Can I use this for non-normal distributions?
A: This formula specifically applies to normal distributions. For other distributions, different methods are needed.