Explanation 1
PMF: p(x)=P(X=x), the mass at one value.
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Lesson 3 of 18
Use PMF and CDF views to answer exact, cumulative, interval, and tail questions.
Core lesson
Use PMF and CDF views to answer exact, cumulative, interval, and tail questions.
Explanation 1
PMF: p(x)=P(X=x), the mass at one value.
Explanation 2
CDF: F(x)=P(X<=x), cumulative probability through x.
Explanation 3
Tail: Probability beyond a threshold.
Terminology
p(x)=P(X=x), the mass at one value.
Exactly 55 requests.
F(x)=P(X<=x), cumulative probability through x.
At most 55 requests.
Probability beyond a threshold.
More than 55 requests.
Notation and formulas
p(x)=P(X=x); F(x)=P(X<=x)
Define X, its support, parameters, units, and assumptions before substituting values.
Interactive Mission
Compare point mass with cumulative probability through a selected x.
P(X = x)
0.2
F(x) = P(X <= x)
0.9
E(X)
3
SD(X)
1.0954
| x | P(X=x) | Included in F(4) |
|---|---|---|
| 1 | 0.1 | Yes |
| 2 | 0.2 | Yes |
| 3 | 0.4 | Yes |
| 4 | 0.2 | Yes |
| 5 | 0.1 | No |
Interpretation: Custom PMF, discrete uniform, Bernoulli, or binomial: inspect support and trial structure before choosing.
Interactive Mission
Translate exactly, at most, fewer than, more than, and at least.
P(X = x)
0.2
F(x) = P(X <= x)
0.9
E(X)
3
SD(X)
1.0954
| x | P(X=x) | Included in F(4) |
|---|---|---|
| 1 | 0.1 | Yes |
| 2 | 0.2 | Yes |
| 3 | 0.4 | Yes |
| 4 | 0.2 | Yes |
| 5 | 0.1 | No |
Interpretation: Custom PMF, discrete uniform, Bernoulli, or binomial: inspect support and trial structure before choosing.
Worked example
Scenario
A desk can handle 55 requests before overtime.
A one-integer boundary error changes the decision probability.
R connection
Run and edit the original script, inspect intermediate output, and keep exact and approximate results clearly labelled.
Live R Lab
Build PMF and CDF tables and answer exact, cumulative, and tail queries.
Ready to run
Dataset provenance
Original synthetic data generated for STATLAB Academy. No textbook data used.
Interpretation safeguard
For integer X, fewer than x is P(X<=x-1), while at least x is 1-P(X<=x-1).
Reflection
Name the random variable, support, parameters, units, dependence assumptions, and one limitation.
Assistant
Coach me through Probability Mass Functions and Cumulative Distribution Functions by asking about X, support, parameters, assumptions, probability notation, exact versus approximate status, and interpretation. Do not provide a quiz answer before submission or write my final capstone recommendation.
Exit check
Use conceptual hints before submitting. Full formulas, calculations, and interpretation appear afterward.
Checkpoint
Question 1 of 2. Answered 0/2. Passing score: 70%.
Resource
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