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How to build the distribution function F(x) from a probability table

The recipe: probabilities must sum to 1, then accumulate

A discrete random variable is given by a table: the values it takes and the probability of each. Two facts do all the work. First, the probabilities of the whole table add up to $1$ - that is what lets you recover a missing entry. Second, the distribution function accumulates them from left to right, so between two neighbouring values of $X$ it is constant, and at each value it jumps by exactly the probability of that value. The graph is therefore a staircase that starts at $0$ and ends at $1$ . One warning about conventions: this textbook defines $F (x) = P (X < x)$ with a strict inequality. Many English-language courses define $F (x) = P (X \leq x)$ instead; the staircase is the same, only the endpoints of each step change from open to closed.

Problem

A random variable $X$ is given by the distribution table

X P 2 0.2 3 ? 4 0.4

Find the distribution function of $X$ and draw its graph.

The graph of F(x): a staircase that jumps by the probability of each value and stays at 1 afterwards.

Answer

p = 0.4

, and

F (x) = ⎩ ⎨ ⎧ 0, 0.2, 0.6, 1, x \leq 2 2 < x \leq 3 3 < x \leq 4 x > 4

Checked by hand: the missing probability and all four steps are correct for the convention F(x) = P(X < x).

Step-by-step solution

1. Recover the missing probability

The probabilities in the table must add up to $1$ . Call the unknown one $p$ :

0.2 + p + 0.4 = 1

p = 1 - 0.6 = 0.4

So $P (X = 3) = 0.4$ .

2. Accumulate the probabilities

The distribution function collects everything strictly to the left of $x$ , that is $F (x) = P (X < x)$ . Walk along the axis: For $x \leq 2$ no value of $X$ is smaller than $x$ , so

F (x) = 0

For $2 < x \leq 3$ only the value $X = 2$ is to the left:

F (x) = 0.2

For $3 < x \leq 4$ the values $2$ and $3$ are to the left:

F (x) = 0.2 + 0.4 = 0.6

For $x > 4$ all three values are to the left:

F (x) = 0.2 + 0.4 + 0.4 = 1

3. Write it as one formula

F (x) = ⎩ ⎨ ⎧ 0, 0.2, 0.6, 1, x \leq 2 2 < x \leq 3 3 < x \leq 4 x > 4

4. Draw the staircase

The graph is constant between the values and jumps at $x = 2, 3, 4$ by $0.2$ , $0.4$ and $0.4$ - exactly the probabilities from the table. It starts on the axis at height $0$ and stays at height $1$ to the right of the last value, as every distribution function must.

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