Distributions · Definition

Normal distribution

probabilitydistributions

Before you start you need to know

It is the most used distribution in statistics: it describes quantities that result from adding many small effects, such as measurement errors or heights.

Definition

A normal variable is a continuous variable with density \(f\) shaped like a symmetric bell, determined by its mean \(\mu\) and its standard deviation \(\sigma\).

\[f(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]

A normal variable is written \(X\sim N(\mu,\sigma^2)\): \(\mu\) is its mean and \(\sigma^2\) its variance, so \(\sigma\) is the standard deviation. The bell is symmetric about \(\mu\), where the density peaks; \(\sigma\) controls the width: the larger \(\sigma\), the lower and wider the bell. When \(Z\sim N(0,1)\) it is called the standard normal.

Every normal reduces to the standard one through \(Z=(X-\mu)/\sigma\). A single table (or a single distribution function, \(\Phi\)) is therefore enough to compute any probability. Moreover, a linear transformation \(aX+b\) of a normal is still normal, with mean \(a\mu+b\) and variance \(a^2\sigma^2\).

The empirical 68-95-99.7 rule summarizes the shape: about 68% of the probability lies within one standard deviation of the mean, 95% within two and 99.7% within three. By symmetry, \(P(Z\le-z)=1-P(Z\le z)\) and \(P(Z\le0)=\tfrac12\).

The density has no elementary antiderivative, so probabilities are read from tables or computed numerically. A common mistake is to confuse the second parameter with the standard deviation: in \(N(5,9)\), \(\sigma=3\). Its theoretical importance comes from the central limit theorem, which explains why it shows up so often.

Think about it

If \(X\sim N(10,4)\), how do you standardize \(P(X\le 13)\)?

With \(\sigma=2\): \(z=(13-10)/2=1.5\), so \(P(X\le13)=P(Z\le1.5)\approx0.933\).

Why can two normals with the same mean look so different?

Because they differ in \(\sigma\): the one with the larger standard deviation is lower and wider, even though both are centered at the same point.

Exercises 10

  1. If \(X\sim N(5,9)\), what is the standard deviation of \(X\)?
  2. If \(X\sim N(50,16)\), what is the standardized value \(z\) of \(x=58\)?
  3. If \(Z\) is standard normal, what is \(P(Z\le0)\)?
  4. If \(X\sim N(\mu,1)\), at which point does the density reach its maximum?
  5. If \(Z\) is standard normal, what is \(f_Z(0)\), the height of the density at 0?
  6. If \(X\sim N(3,4)\), what distribution does \(Y=2X+1\) have?
  7. If \(X\sim N(100,225)\), by the 68-95-99.7 rule, \(P(85\le X\le115)\) is approximately:
  8. If \(Z\) is standard normal, what is \(E[Z^2]\)?
  9. If \(X\sim N(\mu,\sigma^2)\), approximately what is \(P(X\le\mu+\sigma)\)?
  10. Let \(X\sim N(2,\sigma^2)\) with \(P(X>5)=0.1\). What is \(P(X<-1)\)?

What comes next

References

  • Joseph K. Blitzstein, Jessica Hwang, Introduction to Probability, 2nd edition, CRC Press, 2019. §5.4
  • Sheldon Ross, A First Course in Probability, 10th edition, Pearson, 2019. §5.4
  • Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying Ye, Probabilidad y estadística para ingeniería y ciencias, 9.ª edición, Pearson, 2012. §6.3