Sum of normally distributed random variables

Key takeaways

Sum of normally distributed random variables belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Sum of normally distributed random variables to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sum of normally distributed random variables from memory before moving on to harder problems.

Reference excerpt

In probability theory, calculation of the sum of normally distributed random variables is an instance of the arithmetic of random variables. This is not to be confused with the sum of normal distributions which forms a mixture distribution. Addition of random variables, on the other hand, are the convolution of their probability distributions.

Independent random variables Let X and Y be independent random variables that are normally distributed (and therefore also jointly so), then their sum is also normally distributed. i.e., if

X ∼ N ( μ X , σ X 2 ) {\displaystyle X\sim N(\mu _{X},\sigma _{X}^{2})}

Y ∼ N ( μ Y , σ Y 2 ) {\displaystyle Y\sim N(\mu _{Y},\sigma _{Y}^{2})}

Z = X + Y , {\displaystyle Z=X+Y,}

then

Z ∼ N ( μ X + μ Y , σ X 2 + σ Y 2 ) . {\displaystyle Z\sim N(\mu _{X}+\mu _{Y},\sigma _{X}^{2}+\sigma _{Y}^{2}).}

This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations). In order for this result to hold, the assumption that X and Y are independent cannot be dropped, although it can be weakened to the assumption that X and Y are jointly, rather than separately, normally distributed. (See here for an example.) The result about the mean holds in all cases, while the result for the variance requires uncorrelatedness, but not independence.

Proofs

Proof using characteristic functions The characteristic function

φ X + Y ( t ) = E ⁡ ( e i t ( X + Y ) ) {\displaystyle \varphi _{X+Y}(t)=\operatorname {E} \left(e^{it(X+Y)}\right)}

of the sum of two independent random variables X and Y is just the product of the two separate characteristic functions:

φ X ( t ) = E ⁡ ( e i t X ) , φ Y ( t ) = E ⁡ ( e i t Y ) {\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),\qquad \varphi _{Y}(t)=\operatorname {E} \left(e^{itY}\right)}

of X and Y. The characteristic function of the normal distribution with expected value μ and variance σ2 is

φ ( t ) = exp ⁡ ( i t μ − σ 2 t 2 2 ) . {\displaystyle \varphi (t)=\exp \left(it\mu -{\sigma ^{2}t^{2} \over 2}\right).}

So

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Worked examples

Example 1 — a first encounter with Sum of normally distributed random variables

Start with the simplest possible case. Write down what Sum of normally distributed random variables claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Sum of normally distributed random variables before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Sum of normally distributed random variables ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Sum of normally distributed random variables

In research: Sum of normally distributed random variables appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Sum of normally distributed random variables in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Sum of normally distributed random variables is common in secondary-school and first-year university syllabi. It links to neighbouring topics Normal distribution, Theory of probability distributions, so understanding it makes those chapters shorter.
In everyday life: Look for Sum of normally distributed random variables outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Sum of normally distributed random variables in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Sum of normally distributed random variables means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
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Frequently asked questions

What is Sum of normally distributed random variables in simple terms?

In probability theory, calculation of the sum of normally distributed random variables is an instance of the arithmetic of random variables. This is not to be confused with the sum of normal distributions which forms a mixture distribution.

Why does Sum of normally distributed random variables matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Sum of normally distributed random variables?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Sum of normally distributed random variables.