CDF (Cumulative Distribution Function)

The Cumulative Distribution Function (CDF) gives the probability that a random variable X takes a value less than or equal to a certain value x. It is a cumulative measure of probability and is defined for both discrete and continuous random variables.

Definition:

For a random variable X, the CDF is denoted as F_X(x), and it is defined as:

F_X(x) = P(X ≤ x)

Properties of CDF:

Non-decreasing: The CDF never decreases as x increases.

Limits: lim_x→−∞ F_X(x) = 0 and lim_x→∞ F_X(x) = 1

Right-continuous: The CDF is continuous from the right.

For discrete random variables, the CDF is a step function. For continuous random variables, it is a smooth curve.

Example (Discrete):

If X is the outcome of a fair six-sided die, then:

        FX(x) = { 0, x < 1
                        1/6, 1 ≤ x < 2
                        2/6, 2 ≤ x < 3
                        3/6, 3 ≤ x < 4
                        4/6, 4 ≤ x < 5
                        5/6, 5 ≤ x < 6
                        1, x ≥ 6 }
    

Example (Continuous):

For a uniform random variable X over the interval [0,1], the CDF is:

        FX(x) = { 0, x < 0
                        x, 0 ≤ x ≤ 1
                        1, x > 1 }
    

PDF (Probability Density Function)

The Probability Density Function (PDF) describes the likelihood of a continuous random variable X taking on a particular value. Unlike the CDF, the PDF does not give probabilities directly but instead represents the density of the probability at a point.

Definition:

For a continuous random variable X, the PDF f_X(x) is the derivative of the CDF:

f_X(x) = d/dx F_X(x)

Properties of PDF:

Non-negative: f_X(x) ≥ 0 for all x.

Total Area = 1: ∫_−∞^∞ f_X(x) dx = 1

Probability for an interval: The probability that X falls within an interval [a,b] is given by: P(a ≤ X ≤ b) = ∫_a^b f_X(x) dx

Example:

For a uniform random variable X over the interval [0,1], the PDF is:

        fX(x) = { 1, 0 ≤ x ≤ 1
                        0, otherwise }
    

For a standard normal random variable Z with mean 0 and variance 1, the PDF is:

        fZ(z) = 1/√(2π) e−z²/2
    

Relationship Between CDF and PDF

The PDF is the derivative of the CDF: f_X(x) = d/dx F_X(x)

The CDF is the integral of the PDF: F_X(x) = ∫_−∞^x f_X(t) dt

Comparison of CDF and PDF

Aspect	CDF	PDF
Definition	FX(x) = P(X ≤ x)	fX(x) = d/dx FX(x)
Value	Cumulative probability	Probability density
Range	[0,1]	[0,∞)
Graph	S-shaped curve (for continuous random variables)	Bell-shaped or other curves depending on the distribution

Examples

Click the button below to explore examples:

View Examples