Normal

The bell curve that emerges from sums of many independent effects, central to approximation and finance.

1. The idea

A Normal $(μ, σ^{2})$ random variable has density $f (x) = \frac{1}{σ 2 π} e^{- (x - μ)^{2} / (2 σ^{2})}$ and is fully determined by its mean and variance — all higher standardized moments follow from these two.

The Central Limit Theorem is why it appears everywhere: the standardized sum of many independent, identically distributed random variables with finite variance converges to Normal $(0, 1)$ , regardless of the original distribution's shape.

Linear combinations of jointly Normal variables are Normal, which makes the Normal family closed under addition and scaling — essential for reasoning about portfolios and sums of returns.

2. Why it matters

The Normal distribution underlies Binomial and Poisson approximations for large parameters, portfolio return modeling, and confidence-interval reasoning — recognizing when CLT applies turns an intractable sum into a simple $z$ -score calculation.

3. Core mathematics

Parameters

μ \in R, σ^{2} > 0

Support

x \in R

Expectation

E [X] = μ

Variance

Var (X) = σ^{2}

Key property

X \sim N (μ, σ^{2}) \Rightarrow \frac{X - μ}{σ} \sim N (0, 1)

standardization

Sums

X_{i} \sim ii d (μ, σ^{2}) \Rightarrow \frac{\sum X _{i} - n μ}{σ n} d N (0, 1)

Central Limit Theorem

4. How to recognize it

Think of this when…

Large $n$ sums or averages of independent random variables — a CLT approximation cue
"Approximate the probability that a Binomial/Poisson count exceeds $x$ " for large parameters
Portfolio return or P&L modeling described as approximately Normal
Questions asking for $z$ -scores, confidence intervals, or symmetric tail probabilities

5. Worked example

Worked example

A fair coin is flipped $400$ times. Approximate the probability of getting more than $220$ heads.

1
Let $X \sim Binomial (400, 0.5)$ ; exactly $E [X] = 200$ , $Var (X) = 400 (0.5) (0.5) = 100$ .
2
By CLT, approximate $X \approx N (200, 100)$ , so $σ = 10$ .
3
Standardize the threshold.
$z = \frac{220 - 200}{10} = 2$
4
Use the standard normal tail probability.
$P (X > 220) \approx P (Z > 2) \approx 0.0228$
Answer: $P (X > 220) \approx 0.0228$ (about $2.3%$ ).

6. Common mistakes

Forgetting a continuity correction when approximating a discrete distribution (e.g. using $220.5$ for a sharper Binomial approximation)
Applying CLT to a sum of only a few variables, or ones with heavy tails / infinite variance
Confusing $σ$ and $σ^{2}$ when standardizing (dividing by variance instead of standard deviation)
Assuming a sum of Normals is Normal even when the variables are not independent or jointly Normal

7. Quick check

If $X \sim N (50, 16)$ , what is $P (X > 58)$ approximately (using $P (Z > 2) \approx 0.0228$ )?

Why does the Central Limit Theorem justify approximating a Binomial $(n, p)$ for large $n$ with a Normal distribution?

8. Practice

11 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Symmetry

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