The idea, step by step
Mass at a point, total to the left
The probability function gives the chance of one exact value. The cumulative distribution function is . To find F(x), add every probability whose value is no larger than x.
For an integer-valued variable, a strict endpoint changes the sum. For example, P(X < 3) includes 0, 1 and 2, whereas P(X ≤ 3) also includes 3. Translate the wording before using a calculator.
Recover probabilities from the jumps
A discrete cumulative distribution is constant between possible values and rises at them. Its jump at a possible value equals the probability of that value. A cumulative function never falls and eventually reaches 1.
For integer values, and . If the values are not consecutive integers, subtract the cumulative total immediately before the lower included value rather than blindly using a−1.
A common trap
P(X ≥ 2) is 1 − F(2).
Instead: F(2) already includes X = 2. For an integer-valued variable, use 1 − F(1) to retain that endpoint.
See the thinking
Build and use cumulative values
X has probabilities 0.1, 0.3, 0.4, 0.2 at 0, 1, 2, 3. Find F(1), P(X > 1) and P(1 < X ≤ 3).
| x | P(X = x) |
|---|---|
| 0 | 1/10 |
| 1 | 3/10 |
| 2 | 2/5 |
| 3 | 1/5 |
- F(1) includes the values 0 and 1: 0.1 + 0.3 = 0.4.
- P(X > 1) is the complement of X ≤ 1, so it is 1 − 0.4.
- The final event contains values 2 and 3, whose probabilities sum to 0.6.
Read a jump
An integer-valued variable has F(0) = 0.2, F(1) = 0.55 and F(2) = 1. Find P(X = 1) and P(X ≥ 1).
- The jump at 1 is F(1) − F(0) = 0.55 − 0.2.
- The complement of X ≥ 1 is X ≤ 0.
- Therefore P(X ≥ 1) = 1 − F(0).
Check your understanding
If F(2) = 0.7, what is P(X > 2)?
Show answer
0.3.
The events X ≤ 2 and X > 2 are complementary.
Your turn
6 questions · 13 marks · Work at your own pace.
Read the questions online
- For probabilities 0.2, 0.5 and 0.3 at x = 0, 1, 2, give F(0), F(1) and F(2).3 marks
- For the same distribution, find P(X < 2).2 marks
- An integer-valued X has F(2) = 0.65 and F(3) = 0.90. Find P(X = 3).2 marks
- X takes only the values 2, 5 and 8, with probabilities 0.1, 0.6 and 0.3. Find F(6).2 marks
- For integer-valued X, F(1) = 0.3 and F(4) = 0.95. Find P(2 ≤ X ≤ 4).2 marks
- A proposed cumulative function has F(1) = 0.7 and F(2) = 0.6. Explain why it is impossible.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- For probabilities 0.2, 0.5 and 0.3 at x = 0, 1, 2, give F(0), F(1) and F(2).3 marks
Take running totals: 0.2; 0.2 + 0.5; 0.2 + 0.5 + 0.3.
0.2, 0.7, 1. - For the same distribution, find P(X < 2).2 marks
The event includes x = 0 and x = 1 only.
Add 0.2 + 0.5.
0.7. - An integer-valued X has F(2) = 0.65 and F(3) = 0.90. Find P(X = 3).2 marks
Subtract the cumulative total before 3 from the total at 3.
0.90 − 0.65 = 0.25.
0.25. - X takes only the values 2, 5 and 8, with probabilities 0.1, 0.6 and 0.3. Find F(6).2 marks
The values no larger than 6 are 2 and 5.
Add 0.1 + 0.6; there need not be mass at 6.
0.7. - For integer-valued X, F(1) = 0.3 and F(4) = 0.95. Find P(2 ≤ X ≤ 4).2 marks
F(4) includes all values up to 4.
Subtract F(1), which excludes everything below 2: 0.95 − 0.3.
0.65. - A proposed cumulative function has F(1) = 0.7 and F(2) = 0.6. Explain why it is impossible.2 marks
The event X ≤ 1 is contained in X ≤ 2.
Adding possible values cannot reduce total probability.
A cumulative probability cannot decrease.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 Mathematical Methods: discrete random variables and probability distributions, expected value and variance, transformations and independent sums, Bernoulli and binomial models. Original supporting lessons and exercises follow the approved Ascent chapter structure.
Official curriculum source ↗