Distribution Function

• If the discrete random variable X is at least ordinally scaled, we can define the distribution function of X similarly to the empirical distribution function of a feature/characteristic
• For this, we assume that the values \(x_1, x_2, ... , x_k, ...\) in the support of X are already sorted in ascending order (i.e. \(x_1 < x_2 < ... < x_k < ...\))

\[F(x) = P(X \le x) = \sum\limits_{i:x_i \le x} f(x_i) = \sum\limits_{i:x_i \le x} p_i\]

and \(P(X \ge x) = 1 - P(X \le x) = 1 - F(x)\)

• F(x) corresponds exactly to the probability that the value of the random variable X is at most x.

Properties

• The function F is a monotonically increasing step function
• Jump of height \(p_i = f(x_i)\) at \(x_i\)
• At jump points, the upper value (“stair edge”) is the corresponding function value
• F(x) = 0 for all \(x < x_1\)

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For a continuos variable with density f

\[F(x) = P(X \le x) = \int_{-\infty}^{x} f(x) \; dx\]

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For continous uniform distribution

\[F(x) = \int_{- \infty}^{x} f(t) \; dt = \begin{cases} 0 ,& x < a\\ \frac{x - a}{b - a} ,& a \le x \le b \\ 1 ,& x > b \end{cases} \]

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For the exponential distribution

\[F(x) = \begin{cases} 1- e^{-\lambda x} ,& x \ge 0\\ 0 ,& x < 0 \end{cases} \]

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For the poisson distribution

(according to https://de.wikipedia.org/wiki/Poisson-Verteilung#Verteilungsfunktion) \[F(x) = e^{-\lambda}\sum_{k = 0}^{n}\frac{\lambda ^x}{x!}\] —

some calculation hints

Probabilities can be calculated via the distribution function as follows:
- \(P(X > x) = 1 - P(X \le x) = 1 - F(x)\)
- \(P(x < X \le y) = F(y) - F(x)\)
- \(P(x < X < y) = P(X \le y) - P(X \le x) = F(y) - F(x)\)
- \(P(x \le X \le y | X \le z) = \frac{P(x \le X \le y, X \le z)}{P(X \le z)} = \frac {P(x \le X \le z)}{P(X \le z)} = \frac{F(z) - F(x)}{F(z)}\)
- \(P(X \le x) = F(x)\)
- \(P(X \ge x) = 1 - F(x)\)

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