Normal Distribution

All the distributions discussed so far describe a discrete random variable, where every possible value can be listed and assigned its own probability. A continuous random variable — such as height or weight — can take infinitely many values within a range, so it no longer makes sense to assign a probability to each individual value; instead, it is described by a probability density function (PDF).

A continuous random variable $X$ is said to follow a normal distribution (or Gaussian distribution) with parameters mean $\mu = E(X)$ and variance $\sigma^2 = \mathrm{Var}(X)$, written $X \sim N(\mu, \sigma^2)$, if its probability density function is:

where $\mu \in (-\infty, \infty)$ is the mean and $\sigma > 0$ is the standard deviation. The graph of $f(x)$ is the familiar bell-shaped probability curve, with a single peak and tails that extend indefinitely on both sides without ever touching the axis.

A normal distribution has several distinctive features worth remembering:

1. The mean, median and mode all coincide at the same value.
2. The curve has exactly one peak, making the distribution unimodal.
3. The curve is symmetric about $x = \mu$ — exactly half the values lie below the mean and half lie above it.
4. The total area under the curve is always equal to $1$, as required of any probability density function.
5. The entire shape of the distribution is fully described by just two parameters — the mean $\mu$ and the standard deviation $\sigma$. …