I saw that students are unclear about #PDF [probability density function] and #CDF [cumulative density function].
I would be glad to provide a comprehensive explanation of probability density functions (PDFs) and cumulative distribution functions (CDFs), incorporating the best aspects of the previous responses and addressing any potential shortcomings:
Probability Density Function (PDF): A PDF is a mathematical function that describes the probability distribution of a continuous random variable. It represents the likelihood of a random variable taking on a particular value within a certain range.
The PDF is always non-negative and its integral over its entire range must equal 1.
For a continuous random variable X, the PDF is denoted as f(x).
The probability of X falling within a certain range [a, b] is given by the integral of the PDF over that range: P(a ≤ X ≤ b) = ∫[a, b] f(x) dx.
Cumulative Distribution Function (CDF): A CDF is a mathematical function that gives the probability that a random variable is less than or equal to a certain value. It is the integral of the PDF from negative infinity to that value.
For a continuous random variable X, the CDF is denoted as F(x). The CDF is always non-decreasing and its values range from 0 to 1.
The probability of X being less than or equal to a value x is given by F(x): P(X ≤ x) = F(x).
Relationship between PDF and CDF
The PDF is the derivative of the CDF: f(x) = dF(x)/dx.
The CDF is the integral of the PDF: F(x) = ∫[-∞, x] f(t) dt.
Minimal Example
Consider the uniform distribution over the interval [0, 1].
The PDF is: f(x) = 1, 0 ≤ x ≤ 1 f(x) = 0, otherwise The CDF is: F(x) = 0, x lt; 0 F(x) = x, 0 ≤ x ≤ 1 F(x) = 1, x > 1 Key Points PDFs and CDFs are fundamental concepts in probability theory.
PDFs describe the likelihood of a random variable taking on a particular value. CDFs give the probability that a random variable is less than or equal to a certain value.
PDFs and CDFs are related through differentiation and integration.
Another small example of PDF
Given a probability density function, f(x) = 1/100, what is the probability
P(10<X<20), where X~Uniform[0, 100]?
We use the probability density function (PDF) to calculate probabilities over intervals when dealing with continuous random variables.
Since X is uniformly distributed over [0, 100] with f(x) = 1/100,
we calculate P(10 < X < 20) as follows:
P(10 < X < 20) = ∫₁₀²₀ f(x) dx
For a uniform distribution, f(x) = 1/100:
P(10 < X < 20) = ∫₁₀²₀ (1/100) dx = 1/100 × (20 - 10) = 1/100 × 10 = 0.1
Therefore, the probability P(10 < X < 20) is 0.1.
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