Normal distribution flashcards: z-scores & CLT

Question 1

What is a normal distribution?

Accepted Answer

A continuous distribution with density $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right),\ \sigma>0,$$ where $\mu$ is the mean and $\sigma$ is the standard deviation.

Question 2

State the z-score formula for converting $x$ from $N(\mu,\sigma)$ to a standard normal variable

Accepted Answer

$$z=\frac{x-\mu}{\sigma}$$ where $x$ is the original value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Question 3

State the central limit theorem (CLT) for sample means

Accepted Answer

For i.i.d. observations with mean $\mu$ and finite variance $\sigma^2$, the sample mean $\bar X$ is approximately normal for large $n$: $$\bar X\approx N\!\left(\mu,\frac{\sigma^2}{n}\right),$$ regardless of the population distribution.

Normal distribution flashcards: z-scores & CLT

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