Calculate MGF of Sum of Normal Random Variables
Use this interactive calculator to find the moment generating function of a sum of normal random variables, along with the combined mean, variance, standard deviation, and a visual MGF curve based on your chosen value of t and dependence structure.
Expert Guide: How to Calculate the MGF of the Sum of Normal Random Variables
The moment generating function, usually abbreviated as MGF, is one of the most useful tools in probability and mathematical statistics. If you are working with normal random variables, the MGF becomes especially elegant because the normal family is closed under summation. That means the sum of normal random variables is itself normal under very broad conditions, and its MGF can be written in a compact closed form. This calculator is designed to make that process fast, but understanding the logic behind it is equally important for students, data analysts, quants, engineers, and researchers.
Suppose you have random variables X1, X2, …, Xn, where each variable is normally distributed. Define the sum as S = X1 + X2 + … + Xn. The key question is: what is M_S(t), the MGF of the sum? The answer depends on the combined mean and the combined variance. Once you know those two quantities, the MGF follows immediately.
Core result: If S ~ N(μ_S, σ_S²), then its moment generating function is M_S(t) = exp(μ_S t + 0.5 σ_S² t²).
Why the normal distribution makes this easy
The normal distribution is one of the few continuous distributions with a very simple MGF. For a single normal random variable X ~ N(μ, σ²), the MGF is:
M_X(t) = exp(μt + 0.5σ²t²)
This matters because if random variables are independent, the MGF of their sum is the product of their individual MGFs. Multiplying exponentials adds exponents, which gives a new expression that is still of normal form. Even when the variables are correlated, the sum remains normal as long as the joint distribution is multivariate normal. In that case, you simply account for covariance when computing the total variance.
Step by step formula for the sum
- Add all means to get the mean of the sum: μ_S = Σμ_i.
- Add all variances and, if relevant, add covariance terms: σ_S² = Σσ_i² + 2ΣCov(Xi, Xj) for i < j.
- Plug the result into the normal MGF formula: M_S(t) = exp(μ_S t + 0.5 σ_S² t²).
If the variables are independent, every covariance is zero, so the variance formula simplifies to σ_S² = Σσ_i². This is the most common classroom case and is typically the fastest to compute by hand.
Independent vs correlated sums
Many users make a subtle mistake here: they add means correctly but forget that variance is not simply the sum of standard deviations. You must square standard deviations first to get variances. Then, if variables are correlated, you must include covariance terms. Positive covariance increases the spread of the sum. Negative covariance reduces it. The MGF is very sensitive to the variance because variance appears inside the exponential multiplied by 0.5t².
| Case | Mean of sum | Variance of sum | MGF of sum |
|---|---|---|---|
| Independent normal variables | μ_S = Σμ_i | σ_S² = Σσ_i² | exp(μ_S t + 0.5σ_S²t²) |
| Correlated jointly normal variables | μ_S = Σμ_i | σ_S² = Σσ_i² + 2ΣCov(Xi, Xj) | exp(μ_S t + 0.5σ_S²t²) |
Worked example
Take three normal random variables:
- X1 ~ N(2, 1.5²)
- X2 ~ N(5, 2²)
- X3 ~ N(-1, 0.5²)
If they are independent, then:
- Mean: μ_S = 2 + 5 – 1 = 6
- Variance: σ_S² = 1.5² + 2² + 0.5² = 2.25 + 4 + 0.25 = 6.5
- MGF: M_S(t) = exp(6t + 3.25t²)
- At t = 0.4, M_S(0.4) = exp(6(0.4) + 3.25(0.16)) = exp(2.92)
Numerically, that is about 18.54. The calculator above computes that instantly and plots the MGF across a user-selected t range.
What the MGF tells you
The MGF contains all moments of the distribution, when it exists in a neighborhood around zero. For the normal distribution it exists for all real t. Differentiating the MGF and evaluating at zero gives moments. In practice, the MGF confirms the distributional form and makes proofs involving sums much cleaner. In finance, signal processing, quality engineering, and statistical inference, this property is used to derive expectations, variances, and asymptotic approximations.
For normal sums, the MGF also immediately reveals that the resulting distribution remains normal. This is one reason the normal model is foundational in statistics. The Central Limit Theorem explains why approximate normality appears often, but for exact normal variables, the closure under addition is even stronger.
Important interpretation of the chart
The chart produced by this calculator shows how M_S(t) changes as t moves across the selected interval. Every MGF equals 1 at t = 0. Around zero, the slope is influenced by the mean of the sum, and the curvature is influenced by the variance. If the variance is large, the curve grows very quickly as |t| increases. That rapid growth is normal because the MGF contains an exponential quadratic term.
Common mistakes to avoid
- Adding standard deviations directly. You should add variances, not standard deviations.
- Ignoring covariance. If the variables are correlated, your variance of the sum is incomplete without covariance terms.
- Confusing MGF with characteristic function. The MGF uses real t and may not exist for every distribution. The characteristic function always exists, but it is a different object.
- Using non-normal dependence assumptions carelessly. The sum of marginally normal variables is guaranteed to be normal when the joint vector is multivariate normal. Marginal normality alone is not always enough in exotic dependence settings.
Reference statistics for normal distributions
Even though the calculator focuses on MGFs, it helps to remember a few standard normal benchmarks. These values are widely used in statistics, confidence interval construction, and probability approximations.
| Normal benchmark | Approximate probability | Interpretation |
|---|---|---|
| Within 1 standard deviation of the mean | 68.27% | About two-thirds of observations fall in μ ± 1σ |
| Within 2 standard deviations of the mean | 95.45% | Most observations fall in μ ± 2σ |
| Within 3 standard deviations of the mean | 99.73% | Almost all observations fall in μ ± 3σ |
| Two-sided 95% critical z value | 1.96 | Common cutoff for confidence intervals and hypothesis tests |
| Two-sided 99% critical z value | 2.576 | Used for stricter significance thresholds |
How covariance changes the answer
Suppose two normal variables each have variance 4. If they are independent, the variance of the sum is 8. But if their covariance is 1.5, the variance of the sum becomes 4 + 4 + 2(1.5) = 11. That difference materially changes the MGF. At moderate values of t, the correlated case can yield a much larger MGF because the exponential term grows with variance. This is why portfolio theory, measurement systems, and forecasting models all care deeply about covariance structure.
When to use this calculator
- Checking homework or exam preparation in probability theory
- Verifying algebra in a mathematical statistics course
- Modeling additive Gaussian noise in engineering
- Combining multiple normal risk factors in finance
- Studying sums of jointly normal measurements in scientific data analysis
Authoritative references for further study
If you want to verify formulas or deepen your understanding of the normal distribution and related statistical methods, these sources are excellent starting points:
- NIST Engineering Statistics Handbook
- Penn State STAT 414 Probability Theory
- U.S. Census Bureau statistical reference materials
Final takeaway
To calculate the MGF of the sum of normal random variables, you do not need to multiply out complicated expressions term by term. Instead, compute the total mean, compute the total variance including covariance when relevant, and substitute into the normal MGF formula. That single step gives you the exact MGF for the sum. The calculator on this page automates the arithmetic, displays the resulting distribution parameters clearly, and plots the MGF so you can interpret the effect of both mean and variance visually.
If you routinely work with Gaussian models, mastering this one formula will save time and prevent mistakes. It is a compact result with broad usefulness across probability, data science, econometrics, quantitative finance, and applied research.