Formulae and properties

• The general form of its probability density function (PDF) is given by:

$$ f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2} $$

• Continuous probability distribution for a real-valued random variable (i.e. $X\in\mathbb{R}$)
• Described by two parameters:
  • Mean $\mu$ (its location)
  • Standard deviation $\sigma$ (its spread)
• Symmetric around the point when $x=\mu$, which is also the mode, the median, and the mean of the distribution
• Unimodal, which means first derivative is positive for $x<\mu$, negative for $x>\mu$, and zero at $x=\mu$

<aside> ⁉️ Challenge 1 From what you have learnt from the previous module Essential Mathematics, derive the first derivative of the PDF of Gaussian distribution. Next, prove that this distribution is unimodal.

• Answer: </aside>

• Sometimes called a “bell curve”

  • However, many other distributions are also bell-shaped, for example Student’s t and logistic distributions

• When a random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$, we denote $X\sim\mathcal{N}(\mu,\sigma^2)$

Area under the curve

• When the random variable is real-valued, the cumulative distribution function (CDF) of a random variable $X$ is equivalent to the probability measure: $F(x) =Pr(X\leq x)$

<aside> ⁉️ Challenge 2 $Pr(X\leq x)$ and $Pr(X<x)$ has no difference in value. Why?

</aside>

• Properties of the CDF:

  • $F(x)$ is non-decreasing and right-continuous
  • $0\leq F(x)\leq 1$, also $\lim\limits_{x\to-\infin}F(x)=0$ and $\lim\limits_{x\to+\infin}F(x)=1$
  • $Pr(a<X<b)=F(b)-F(a)$

  

  $Pr(a<X<b)=F(b)-F(a)$ can be illustrated using this diagram showing the calculation of the area under the normal curve between a lower bound and an upper bound. Whitlock & Schluter (2020). The Analysis of Biological Data, 3e.

• CDF of the normal distribution is given by the integral of the PDF: