Continuous Distributions

Normal Distribution

It's important to note that this is a kind of continuous distribution. Nobody's height is exactly 180 cm, they just fall somewhere between 179 and 181, for example. When we think of the normal distribution, think about infinitesimally small intervals of size $d x$ :

The probability of being in a certain interval is the output of the probability density function. Think of the green bar in the chart above - it's a rectangle with width $d x$ and height $f (x)$ - so $f (x) d x$ gives its area. Therefore, the probability of a value $ξ$ being between $x$ and $x + d x$ is given by:

P (x < ξ < x + d x) = f (x) d x

As you can see on the chart, there is a small part in between the top of the rectangle and the curve. If $d x$ is infinitesimally small, this amount is also infinitesimally small and thus can be ignored.

Probability Density Function

What if we are looking at the probability for a larger area with real, not infinitesimally small numbers? We need to take the integral and do some calculus to add up infinitely many bars of width $d x$ :

P (a < ξ < b) = \int_{a}^{b} f (x) d x

This formula gives us the probability that $ξ$ is between two real numbers $a$ and $b$ .

What about getting the area from $a = - \infty$ and $b = \infty$ ?

\int_{- \infty}^{\infty} f (x) d x = 1

The probability of having a height anywhere in the distribution is 100%.

Normal Dist Function

f (x) = \frac{1}{\sqrt{2 π} σ} e^{\frac{- (x - m)^{2}}{2 σ^{2}}}

Where $m$ is the mean (expectation) and $σ^{2}$ is the variance (not sample variance).

The point where a normal distribution switches from concave to convex is at $m + σ$ (and $m - σ$ )

So, if we take the integral of this function, we can solve for the probability of being in a certain interval. However, there is catch - there is no integral of this formula. But people have manually evaluated this by calculating the area of very small rectangles as we did theoretically above, and compiled these into big tables.

Now we can also use online calculators for this purpose.

Why do we see the normal distribution everywhere?

The reason the normal distribution theorem is the most important distribution of all is beccause of the Central Limit Theorem.