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.


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 < \(x_i\)   \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}= 1\)

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