Answer to Question #257255 in Statistics and Probability for Teddy

Question

Describe the components necessary to explain a distribution.

Answer 1

The first thing that is sufficient to explain a distribution:

Probability сumulative distribution function. F(x) = P(a ≤ x), where a - random variable, x - some value

So, the value of this function represents the probability that a will be less or equal than x(in case of random variable it can be written as P(a < x) cause P(x=a) = 0)

If the random variable is discrete, this function can be derivative into next form:

$P(x=a{\scriptscriptstyle 1}) = p{\scriptscriptstyle 1}$

$P(x=a{\scriptscriptstyle 2}) = p{\scriptscriptstyle 2}$

...

$P(x=a{\scriptscriptstyle n}) = p{\scriptscriptstyle n}$

Where $a{\scriptscriptstyle 1},... a{\scriptscriptstyle n}$ - the values this variable can take. $p{\scriptscriptstyle 1}+...+p{\scriptscriptstyle n}$ = 1.

This is the second thing that is sufficient to explain a distribution. But the first way can be uniquely defined from second and vice verse, so it is sufficient to know just one of them.

In case of continuous random variable the amount of values it can take is uncountable, so for such variables second way defined from the first way using derivative:

$f(x) = F'(x)$ , f(x) is known as probability density function. In that case ${\frac {f(a)} {f(b)}}$ represents the probability ratio for variable of getting into the area a to getting into the area b.

The integral of f(x) over the entire domain is equal to one.

In the case of continuous variable this two ways of explaining it also can be uniquely defined one from another, so it is sufficient to know just one of them