ProbabilityConcept reference

Cumulative distributions

The CDF F(t) is the probability that X is at most t.

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On this page 7 sections

Overview
Understand the idea
Between two possible values, the cumulative probability stays the same
A point mass p(t) asks for exactly t
Key takeaway
Sources & further reading
Concept connections

01Understand the idea#

The cumulative distribution function, or CDF, asks how much probability has accumulated up to a threshold. The endpoint is included: $F(t)=P(X\le t)$ .

Figure 1The CDF is 0 below −1, .2 from −1 through inputs below 2, .7 from 2 through inputs below 4, and 1 from 4 onward. Filled endpoints include each jump; open endpoints exclude the previous height. F(2)=.7.

Here the masses are $0.2$ at $-1$ , $0.5$ at $2$ , and $0.3$ at $4$ . At threshold $2$ , include the first two values. Their masses add to $0.2+0.5=0.7$ .

F(2)=0.7

Check your reasoning

PMF: $p(-2)=0.2$ , $p(0)=0.4$ , $p(3)=0.4$ . Find $F(0)=P(X\le 0)$ .

Show answer and explanation

0.6

Include $-2, 0$ : $0.2+0.4=0.6$ .

02Between two possible values, the cumulative probability stays the same#

Between two possible values, the cumulative probability stays the same. No new mass is crossed.

For this model, $F(3)=F(2)=0.7$ . Below $-1$ , the total is zero. At $4$ or above, it is one.

Check your reasoning

A count $X$ has pmf: $p(0)=0.3$ , $p(2)=0.5$ , $p(5)=0.2$ . Find the probability of at most $4$ .

Show answer and explanation

0.8

Include $0, 2$ : $0.3+0.5=0.8$ .

03A point mass p(t) asks for exactly t#

A point mass $p(t)$ asks for exactly $t$ . The CDF $F(t)$ also includes every possible value below $t$ .

When the threshold equals a possible value, include its mass. That is where a discrete CDF jumps.

Check your reasoning

PMF: $p(-2)=0.1$ , $p(1)=0.7$ , $p(4)=0.2$ . A learner uses only $p(1)$ for $F(1)$ . Repair $F(1)$ .

Show answer and explanation

0.8

Include $-2, 1$ : $0.1+0.7=0.8$ .

Key takeaway

Include every mass at or below the threshold.

Compute a discrete cumulative distribution at a threshold.

Sources & further reading

[1]
Pishro-Nik, Introduction to Probability: Cumulative Distribution Function ↗Pishro-Nik: Introduction to Probability · Article

01Understand the idea#

02Between two possible values, the cumulative probability stays the same#

03A point mass p(t) asks for exactly t#

Include every mass at or below the threshold.

Sources & further reading

Reference this concept