Probability Density Function Calculator
PDF Calculator
Calculate Probability Density Function (PDF) values for continuous distributions with step-by-step solutions and graphical visualizations.
PDF Calculation Result
0.3989
PDF Calculation:
PDF Visualization:
Distribution Properties:
PDF Interpretation:
Probability Density Function (PDF) gives the relative likelihood of a continuous random variable at a specific point.
What is Probability Density Function (PDF)?
Probability Density Function (PDF) is a function that describes the relative likelihood for a continuous random variable to take on a given value. Unlike probability mass functions for discrete variables, PDF gives density rather than probability at a point. The probability of the variable falling within a particular range is given by the integral of the PDF over that range, with the total area under the PDF curve always equal to 1.
Common PDF Formulas
Normal Distribution
Bell curve
Most common
Exponential Distribution
Memoryless
Waiting times
Uniform Distribution
Constant density
Equal likelihood
Properties
Total area = 1
Non-negative
Key PDF Formulas and Properties
1. Normal (Gaussian) Distribution
Where:
μ = mean (center of distribution)
σ = standard deviation (spread)
σ² = variance
Domain: -∞ < x < ∞
2. Exponential Distribution
f(x) = 0 for x < 0
Mean: 1/λ, Variance: 1/λ²
Memoryless property: P(X > s+t | X > s) = P(X > t)
3. Uniform Distribution
| Property | Formula | Interpretation | Example [0,1] |
|---|---|---|---|
| f(x) = 1/(b-a) for a≤x≤b | Constant density | f(x) = 1 | |
| Mean | μ = (a+b)/2 | Center point | 0.5 |
| Variance | σ² = (b-a)²/12 | Spread measure | 1/12 ≈ 0.0833 |
| CDF | F(x) = (x-a)/(b-a) | Cumulative probability | F(x) = x |
Real-World Applications
Statistics & Data Science
- Statistical Modeling: Fitting distributions to data
- Hypothesis Testing: Calculating p-values and test statistics
- Maximum Likelihood Estimation: Parameter estimation using PDFs
- Bayesian Statistics: Prior and posterior distributions
Engineering & Physics
- Signal Processing: Noise modeling and analysis
- Reliability Engineering: Failure time distributions
- Quantum Mechanics: Wave function probability densities
- Thermodynamics: Molecular speed distributions
Finance & Economics
- Risk Management: Modeling asset returns and losses
- Option Pricing: Black-Scholes model assumptions
- Economic Forecasting: Probability distributions of economic variables
- Actuarial Science: Insurance claim distributions
Machine Learning & AI
- Generative Models: Learning data distributions
- Anomaly Detection: Identifying outliers using PDFs
- Bayesian Networks: Probabilistic graphical models
- Reinforcement Learning: Policy gradient methods
Common PDF Examples and Values
| Distribution | Parameters | X value | PDF f(x) | Interpretation |
|---|---|---|---|---|
| Standard Normal | μ=0, σ=1 | 0 | 0.3989 | Highest point (mode) |
| Standard Normal | μ=0, σ=1 | 1 | 0.2420 | 1 standard deviation from mean |
| Exponential | λ=1 | 0 | 1.0000 | Maximum value |
| Exponential | λ=1 | 1 | 0.3679 | Mean waiting time |
| Uniform | [0,1] | 0.5 | 1.0000 | Constant throughout interval |
| Normal | μ=100, σ=15 | 100 | 0.0266 | IQ distribution mean |
Step-by-Step PDF Calculation Process
Example 1: Standard Normal PDF at x=0
- Parameters: μ = 0, σ = 1, σ² = 1
- PDF formula: f(x) = (1/√(2πσ²)) × e^(-(x-μ)²/(2σ²))
- Substitute values: f(0) = (1/√(2π×1)) × e^(-(0-0)²/(2×1))
- Simplify exponent: (0-0)² = 0, so e^0 = 1
- Calculate constant: 1/√(2π) ≈ 1/2.5066 ≈ 0.3989
- Final result: f(0) = 0.3989
- Interpretation: At the mean, standard normal PDF has its maximum value
Example 2: Exponential PDF (λ=1) at x=1
- Parameters: λ = 1, x = 1
- PDF formula: f(x) = λe^(-λx) for x ≥ 0
- Substitute values: f(1) = 1 × e^(-1×1) = e^(-1)
- Calculate: e^(-1) ≈ 0.3679
- Interpretation: Probability density at the mean waiting time (1/λ = 1)
- Note: Area under curve from 0 to ∞ equals 1
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Frequently Asked Questions (FAQs)
Q: What's the difference between PDF and probability?
A: PDF gives density at a point, not probability. For continuous variables, probability at a single point is zero. Probability over an interval [a,b] is ∫[a,b] f(x)dx. PDF values can be greater than 1, but the area under the entire curve always equals 1.
Q: Can PDF values be greater than 1?
A: Yes, PDF values can be greater than 1. What matters is that the total area under the PDF curve equals 1. For example, a uniform distribution on [0, 0.1] has PDF = 10 for all x in [0, 0.1], but area = 10 × 0.1 = 1.
Q: How do I convert PDF to probability?
A: To get probability P(a ≤ X ≤ b), integrate the PDF from a to b: P = ∫[a,b] f(x)dx. For the normal distribution, this requires numerical integration or using the CDF (cumulative distribution function).
Q: What is the relationship between PDF and CDF?
A: CDF F(x) = P(X ≤ x) = ∫[-∞,x] f(t)dt. PDF is the derivative of CDF: f(x) = dF(x)/dx. CDF gives probabilities, PDF gives density rates of change.
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