Continuous Variables

A continuous random variable can take on any real value. This means that even in a finite range of possible values, say from 0 to 1, that there is an infinite number of possible values.

Because there is an infinite range of possible values there is zero probability that a variable takes on an exact value…! Rather we can only describe the probability that the variable falls within a range of values. Yes, this should make your head hurt. The probability of a continuous variable is best described by a probability density function or probability function. An example would be the function shown below.

In general PDF's (for lack of a better acronym) have one or more peaks indicating the most likely values that the variable can take on. The height or value of the function indicates probability (density). The area under the curve represents the actual probability, thus the total area under the curve must be 1. In fancier math classes you may have to force the area to be 1 by "normalizing" the function.

Thus for a given probability density function f(x) the probability that the variable X will be between c and d is given by the integral:

(1)
\begin{align} \int_c ^d f(x) dx = P(c\leq X \leq d) \end{align}

The mean or expected value for a PDF is given by:

(2)
\begin{align} E(X)=\mu = \int_a ^b x f(x) dx \end{align}

Where a and b are the max and min possible values (which can be infinite).

#### Normal Distribution

Many continuous random variables are described by well known probability density functions. One of the most common or at least most important (certainly according to the IB) is the Normal Distribution. The Normal Distribution is symmetric around a mean value and has no limits in terms of max or min values for the variable.

The normal distribution does a good job of modeling things such as heights, weights, IB test scores, etc. With the notable exception that there is a minimum height and often a maximum score on a test… Whereas the normal distribution has no such limitations.

A normal distribution is given by the equation:

(3)
\begin{align} f(x)=\frac{1}{ \sigma \sqrt{2 \pi}} e^{-\frac{1}{2} \left ( \frac{x-\mu}{\sigma} \right ) ^2} \end{align}

Where $\mu$ is the mean of the distribution and $\sigma ^2$ is the variance thus making $\sigma$ the standard deviation. The value of the mean simply shifts where the peak is in the distribution and the standard deviation affects the spread or how wide the distribution is. The smaller the $\sigma$ the taller and narrower the peak.

#### Properties of the Normal Distribution

A normal distribution takes on a "bell curve" shape as shown in the graph at the top of the page and is symmetrical around the mean value. Its worth noting that the value of the probability density function at the mean is:

(4)
\begin{align} f(x=\mu)=\frac{1}{ \sigma \sqrt{2 \pi}} \end{align}

Thus the coordinates of peak of a normal distribution is $(\mu, \frac{1}{ \sigma \sqrt{2 \pi}})$. I would guess that info could come in handy.

The standard deviation also has a special role geometrically. It turns out that if you differentiate the probability density function twice and set it equal to zero, thus finding the inflection points, that the inflection points occurs when:

(5)
\begin{align} x=\mu \pm \sigma \end{align}

In other words, the $\sigma$ is the horizontal distance from the mean of the distribution to the inflection points!

People often refer to "how many sigma's" a value is from the mean. The diagram below shows the distribution of probability as a function of "sigma's." Notice that approximately 68% of the values fall within one sigma of the mean. While roughly only 4% fall more than 2 sigma's from the mean!