# Distribution Function of a Random Variable

## Distribution Function of a Random Variable

Probability that the values $$x_i$$ taken by the random variable X will be strictly less than a given value.

F($$x_i$$) = P(X ≤ $$x_i$$)

A distribution function is a step function that is zero for all values $$x_i$$ less than or equal to the least value of X, and equal to one for all values $$x_i$$ strictly greater than the greatest value of X.

### Example

Consider an honest die with six faces that are identified by the Ace, King, Queen, Jack, and the numbers ten and two from a deck of cards. Consider a random variable defined by the function X from the set Ω of possible results on a set E of values attributed to these faces such as E = {100, 50, 20, 10, 5, 1}, as represented in this table:

 Result (A) P(A) X(A) Ace $$\frac{1}{6}$$ 100 King $$\frac{1}{6}$$ 50 Queen $$\frac{1}{6}$$ 20 Jack $$\frac{1}{6}$$ 10 Ten $$\frac{1}{6}$$ 5 Two $$\frac{1}{6}$$ 1

This means that a player earns 50 points if they draw a King, or one point if they draw a Two. The gains are distributed in the interval [1, 100].

The function F is the distribution function of the random variable X and F($$x_i$$), where $$x_i$$ ∈ [1, 100], is the probability of the event “the value of the random variable X is strictly less than $$x_i$$“.

The values of F are described in this table:

 $$x_i$$ F($$x_i$$) = P(X < $$x_i$$) $$x_i$$ ≤ 1 0 1 < $$x_i$$ ≤ 5 $$\frac{1}{6}$$ 5 < $$x_i$$ ≤ 10 $$\frac{1}{6}+\frac{1}{6}=\frac{1}{3}$$ 10 < $$x_i$$ ≤ 20 $$\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{1}{2}$$ 20 < $$x_i$$ ≤ 50 $$\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{2}{3}$$ 50 < $$x_i$$ ≤ 100 $$\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{5}{6}$$ 100 < $$x_i$$ $$\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}= 1$$