Cumulative Distribution Function (CDF) Calculator
CDF Calculation Tool
Calculate cumulative distribution functions for normal, binomial, exponential, and other probability distributions.
CDF Calculation Result
Probability Distribution
Cumulative Distribution Function
Quantile Calculation
Calculation Details
Distribution: Normal(0, 1)
Formula Used: F(x) = Φ((x-μ)/σ)
Z-score: z = 1.96
Calculation: Φ(1.96) = 0.9750
Interpretation: 97.5% of values fall below x = 1.96
The Cumulative Distribution Function F(x) gives the probability that a random variable X takes on a value less than or equal to x. For the standard normal distribution, F(1.96) = 0.9750 means 97.5% of values fall below 1.96 standard deviations.
What is Cumulative Distribution Function (CDF)?
The Cumulative Distribution Function (CDF) of a random variable X, denoted F(x), is the probability that X takes a value less than or equal to x. It completely characterizes the probability distribution and is fundamental to statistical analysis, hypothesis testing, and probability theory.
Properties of CDF
Monotonic
Always non-decreasing
Never decreases with x
Bounds
Probability values
Between 0 and 1
Limits
Approaches bounds
Right-continuous
Continuous from right
May have jumps
CDF Formulas for Common Distributions
1. Normal Distribution N(μ, σ²)
Where Φ(z) is the standard normal CDF:
- No closed form: Requires numerical approximation
- Symmetry: Φ(-z) = 1 - Φ(z)
- Common values: Φ(0) = 0.5, Φ(1.96) ≈ 0.975, Φ(2.576) ≈ 0.995
- Applications: Statistical testing, quality control, natural phenomena
2. Binomial Distribution B(n, p)
Properties:
- Discrete distribution: Defined for integer k = 0, 1, ..., n
- Step function: Constant between integers, jumps at integers
- Normal approximation: For large n, approximate with normal distribution
- Applications: Success/failure experiments, quality control, survey analysis
3. Exponential Distribution Exp(λ)
Properties:
- Memoryless: P(X > s + t | X > s) = P(X > t)
- Mean: 1/λ, Variance: 1/λ²
- Related to Poisson: Time between events in Poisson process
- Applications: Survival analysis, reliability engineering, queueing theory
4. Uniform Distribution U(a, b)
Properties:
- Linear CDF: Straight line between a and b
- Constant PDF: f(x) = 1/(b-a) for a ≤ x ≤ b
- Maximum entropy: Most uncertain distribution given bounds
- Applications: Random number generation, Monte Carlo methods, prior distributions
CDF vs PDF Comparison
| Aspect | Cumulative Distribution Function (CDF) | Probability Density Function (PDF) | Relationship |
|---|---|---|---|
| Definition | F(x) = P(X ≤ x) | f(x) = dF(x)/dx | CDF is integral of PDF |
| Range | 0 to 1 (probability) | ≥ 0 (not probability) | PDF can be > 1 |
| Interpretation | Cumulative probability up to x | Probability density at x | Area under PDF = CDF |
| Discrete Case | Step function | Probability mass function (PMF) | CDF sums PMF |
| Continuous Case | Continuous function | Continuous function | F(x) = ∫f(t)dt |
| Use Cases | Hypothesis testing, quantiles, probabilities | Likelihood, mode, distribution shape | Complementary information |
Common CDF Values and Interpretations
Standard Normal Distribution Φ(z)
| z-score | CDF Φ(z) | Interpretation | Application |
|---|---|---|---|
| -3.00 | 0.0013 | 0.13% below | Extreme outliers |
| -2.00 | 0.0228 | 2.28% below | 95% confidence lower bound |
| -1.96 | 0.0250 | 2.5% below | Two-tailed test α=0.05 |
| -1.00 | 0.1587 | 15.87% below | One standard deviation below |
| 0.00 | 0.5000 | 50% below | Median |
| 1.00 | 0.8413 | 84.13% below | One standard deviation above |
| 1.96 | 0.9750 | 97.5% below | Two-tailed test α=0.05 |
| 2.00 | 0.9772 | 97.72% below | 95% confidence upper bound |
| 3.00 | 0.9987 | 99.87% below | Extreme outliers |
Step-by-Step CDF Calculation Examples
Example 1: Standard Normal CDF at z = 1.96
- We want to calculate Φ(1.96) = P(Z ≤ 1.96)
- Standard normal distribution: μ = 0, σ = 1
- Since no closed form, use numerical approximation or table lookup
- Result: Φ(1.96) ≈ 0.975002
- Interpretation: 97.5% of values in a standard normal distribution fall below 1.96
- Complement: P(Z > 1.96) = 1 - 0.9750 = 0.0250 (2.5% in right tail)
- Symmetry: Φ(-1.96) = 1 - Φ(1.96) = 0.0250
Example 2: Binomial CDF for n=10, p=0.5, k=5
- We want P(X ≤ 5) where X ~ Binomial(10, 0.5)
- Calculate individual probabilities using PMF: P(X = i) = C(10,i) × 0.5¹⁰
- P(X = 0) = C(10,0) × 0.5¹⁰ = 1 × 0.0009766 = 0.0009766
- P(X = 1) = C(10,1) × 0.5¹⁰ = 10 × 0.0009766 = 0.0097656
- P(X = 2) = 0.0439453, P(X = 3) = 0.1171875, P(X = 4) = 0.2050781, P(X = 5) = 0.2460938
- Sum probabilities: 0.0009766 + 0.0097656 + 0.0439453 + 0.1171875 + 0.2050781 + 0.2460938 = 0.6230469
- Result: P(X ≤ 5) ≈ 0.6230
- Interpretation: There's a 62.3% chance of getting 5 or fewer successes in 10 trials with 50% success probability
Applications of CDF in Statistics
Hypothesis Testing
- P-values: CDF used to calculate tail probabilities
- Critical values: Inverse CDF gives rejection regions
- Test statistics: Compare observed statistic to null distribution CDF
- Confidence intervals: Based on quantiles from CDF
Statistical Inference
- Parameter estimation: Maximum likelihood using CDF
- Goodness-of-fit tests: Kolmogorov-Smirnov test compares empirical CDF to theoretical
- Quantile estimation: Percentiles, quartiles, median from CDF
- Risk analysis: Value at Risk (VaR) calculations
Probability Theory
- Distribution characterization: CDF uniquely defines distribution
- Convergence: Weak convergence defined via CDF
- Transformations: CDF of transformed variables
- Order statistics: Distribution of minimum, maximum, median
Engineering & Science
- Reliability engineering: Survival analysis using CDF
- Signal processing: Probability of signal exceeding threshold
- Quality control: Process capability indices
- Financial modeling: Option pricing, risk measures
Related Functions and Concepts
1. Survival Function S(x)
Applications: Reliability analysis, survival analysis, right-censored data
2. Hazard Function h(x)
Interpretation: Instantaneous failure rate given survival up to time x
3. Quantile Function Q(p)
Also called: Inverse CDF, percent-point function
4. Probability Plot
Graphical method comparing empirical CDF to theoretical CDF
Related Calculators
Frequently Asked Questions (FAQs)
Q: What's the difference between CDF and PDF?
A: CDF gives cumulative probability (area under PDF up to x), while PDF gives probability density (height of distribution at x). CDF ranges 0-1, PDF can be any non-negative value.
Q: How do I calculate CDF for a normal distribution?
A: For X ~ N(μ, σ²), first compute z-score: z = (x-μ)/σ. Then find Φ(z) using standard normal CDF table, statistical software, or numerical approximation (like our calculator!).
Q: Can CDF be greater than 1?
A: No! By definition, CDF is a probability, so 0 ≤ F(x) ≤ 1 for all x. It approaches 0 as x→-∞ and approaches 1 as x→∞.
Q: What is the relationship between CDF and quantiles?
A: Quantiles are the inverse of CDF. The p-th quantile Q(p) satisfies F(Q(p)) = p. For example, the median is Q(0.5) where F(median) = 0.5.
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