1. What is Probability of Error Calculator?

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

Purpose: It helps in statistical analysis, signal processing, and communication systems to determine error probabilities.

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 (tail probability of standard normal distribution)

Explanation: The Q-function calculates the probability that a standard normal random variable will obtain a value larger than x.

3. Importance of Probability of Error Calculation

Details: This calculation is crucial in digital communications, hypothesis testing, and any scenario where you need to determine the likelihood of misclassification between two distributions.

4. Using the Calculator

Tips: Enter the means of both distributions and the standard deviation. The standard deviation must be greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is the Q-function?
A: The Q-function is the tail probability of the standard normal distribution, defined as Q(x) = 1 - Φ(x), where Φ is the cumulative distribution function.

Q2: What does the probability of error represent?
A: It represents the likelihood that a value from one distribution will be mistakenly classified as coming from the other distribution.

Q3: When would I use this calculation?
A: Common applications include digital communication systems, statistical hypothesis testing, and pattern recognition.

Q4: How accurate is the Q-function approximation?
A: The approximation used is accurate to within 1.5×10−7 for all real x.

Q5: Can I use this for non-normal distributions?
A: The formula assumes normal distributions. For other distributions, different methods would be needed.