Question

How do I calculate the probability that the sum of one independent normal distribution is greater than the sum of another, given their means and variances?

Answer 1

To calculate the probability that the sum of one independent normal distribution is greater than the sum of another, given their means and variances, you can follow these steps:

### Step 1: Define the Distributions

Assume you have two independent normal distributions:
- \( X \sim N(\mu_1, \sigma_1^2) \)
- \( Y \sim N(\mu_2, \sigma_2^2) \)

### Step 2: Define the Problem

You want to find the probability \( P(X > Y) \).

### Step 3: Define the New Variable

Since \( X \) and \( Y \) are independent, you can create a new variable \( Z \) defined as:

\[
Z = X - Y
\]

### Step 4: Determine the Distribution of \( Z \)

Next, you need to determine the distribution of \( Z \). Since \( X \) and \( Y \) are independent normal variables:

1. The mean of \( Z \):
\[
\mu_Z = \mu_1 - \mu_2
\]

2. The variance of \( Z \):
\[
\sigma_Z^2 = \sigma_1^2 + \sigma_2^2
\]

Thus, the distribution of \( Z \) is:
\[
Z \sim N(\mu_Z, \sigma_Z^2) = N(\mu_1 - \mu_2, \sigma_1^2 + \sigma_2^2)
\]

### Step 5: Calculate the Probability

Now, you want to find:
\[
P(X > Y) = P(Z > 0)
\]

Using the cumulative distribution function (CDF) of the normal distribution, this can be calculated as:
\[
P(Z > 0) = 1 - P(Z \leq 0) = 1 - \Phi\left(\frac{0 - \mu_Z}{\sigma_Z}\right)
\]
Where \( \Phi \) is the CDF of the standard normal distribution.

### Step 6: Standardize

To use the standard normal distribution, standardize the variable:
\[
\frac{0 - \mu_Z}{\sigma_Z} = \frac{0 - (\mu_1 - \mu_2)}{\sqrt{\sigma_1^2 + \sigma_2^2}} = \frac{\mu_2 - \mu_1}{\sqrt{\sigma_1^2 + \sigma_2^2}}
\]

### Final Step: Calculate the Probability

Thus, the final expression for the probability is:
\[
P(X > Y) = 1 - \Phi\left(\frac{\mu_2 - \mu_1}{\sqrt{\sigma_1^2 + \sigma_2^2}}\right)
\]

You can use standard normal distribution tables or computational tools (like Python's SciPy library) to find the value of \( \Phi \) and compute the desired probability.