What Is CDF Probability?
The cumulative distribution function, F(x) = P(X ≤ x), is enough to find any interval probability. A lower tail is F(a), an upper tail is 1 − F(a), and a range is F(b) − F(a).
Pick the distribution and its parameters, choose the kind of probability, and enter a and b. The calculator shows the formula it used, the result as a decimal and a percentage, and the shaded region.
For discrete distributions such as the binomial and Poisson, the end points are included, and the calculator handles the one-step shift for you.
How to Use the CDF Probability Calculator
- Choose the distribution and enter its parameters.
- Pick the type of probability and enter a, and b if needed.
- Read the probability and check the shaded graph.
CDF Probability Formula
Range
Upper tail
CDF Probability Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| F(x) | Cumulative distribution function | P(X ≤ x) |
| a, b | The cut-off values | |
| F(b) − F(a) | Probability between a and b | |
| 1 − F(a) | Upper tail probability |
CDF Probability Worked Example
With the example values loaded in the calculator above, the steps are:
- Distribution: Normal(μ = 50, σ = 10).
- Write the probability using the CDF F(x) = P(X ≤ x): P(40 < X < 65) = F(65) − F(40).
- Evaluate: 0.774538.
Result: P(40 < X < 65) = 0.774538; As a percentage = 77.454%.
Worked example: Normal(50, 10)
For P(40 < X < 65), the z scores are −1 and 1.5. F(65) = 0.9332 and F(40) = 0.1587.
The probability is 0.9332 − 0.1587 = 0.7745.
Discrete distributions
For a binomial or Poisson count, P(a ≤ X ≤ b) = F(b) − F(a − 1), because the CDF at a already includes a.
Where CDF Probability Is Used
Probability homework
Find less than, greater than and between probabilities.
Normal and t problems
Get the area in a range without tables.
Binomial and Poisson counts
Add up a run of counts in one step.
Common CDF Probability Mistakes to Avoid
- Draw the area before choosing the formula.
- Check that your tails and middle add to 1.
CDF Probability Terms and Definitions
- CDF
- F(x) = P(X ≤ x).
- Tail
- The extreme region of a distribution beyond a cut-off.
CDF Probability FAQs
How do you find a probability from a CDF?
For P(X ≤ a) read F(a). For P(X > a) use 1 − F(a). For a range, subtract the CDF values.
How do I find P(a < X < b)?
Compute F(b) − F(a).
Does it matter if I use < or ≤?
Not for continuous distributions. For discrete ones, it does.
What is the probability outside two values?
1 − [F(b) − F(a)].