Question

make a list of distributions that you can count on to be continuous.

Your examples should be real life examples, rather than general characteristics of continuous variables.

Answer 1

What is a continuous distribution?

A continuous distribution describes the probabilities of the possible values of a continuous random variable. A continuous random variable is a random variable with a set of possible values (known as the range) that is infinite and uncountable.

The list of some continuous distributions are:

note: cauchy distribution defines unpredectibality.

A common conceit for doing this is to consider a blindfolded archer trying to hit a target directly in front of him. He randomly shoots towards the wall at an angle θθ that can sometimes be so large he shoots parallel to the wall!

The archer is standing at the point (0, 0). The point on the wall directly in front of him is (x, 0), and the arrow will land at (x, y), (x, -y), or not at all. After changing to polar coordinates, a moments reflection will give you the equation y=xtan(θ)y=xtan(θ).

Assuming that theta is uniformly distributed on the interval I=(−π/2,π/2)I=(−π/2,π/2), a direct substitution into the equation for the CDF of the uniform distribution will yield the CDF for the Cauchy distribution.

P(Y≤y)=P(xtan(θ)≤y)=P(θ≤arctan(y/x))=arctan(y/x)/π+1/2P(Y≤y)=P(xtan(θ)≤y)=P(θ≤arctan(y/x))=arctan(y/x)/π+1/2

Differentiating this gives the Cauchy density function:f(y)=xπ(x2+y2)

(This example is followed from a source)