Distribution Functions and Densities

Definition

A distribution function of a random variable $X$, also called the cumulative distribution function (CDF), is defined as

$$F(x) = P(X \le x)$$

for every real number $x$.

A density function, more precisely a probability density function (PDF) for a continuous random variable $X$, is a function $f(x)$ such that the probability that $X$ lies in an interval $[a,b]$ is

$$P(a \le X \le b) = \int_a^b f(x)\,dx$$

and the distribution function is obtained by

$$F(x) = \int_{-\infty}^{x} f(t)\,dt$$

for all $x$.

In simple terms:

• The distribution function tells the probability that the random variable has reached or passed a value.
• The density function tells how probability is distributed over the number line for continuous variables.

Main Content

1. Distribution Function

Meaning and interpretation

• The distribution function $F(x)=P(X\le x)$ gives the cumulative probability up to $x$.
• It answers “What is the probability that the random variable is at most $x$?”
• It is used for both discrete and continuous random variables.
• Example: If $X$ represents the waiting time for a bus, then $F(10)=0.75$ means there is a 75% chance the bus arrives within 10 minutes.

Important properties

• $F(x)$ is always between 0 and 1.
• It is non-decreasing: if $a<b$, then $F(a)\le F(b)$.
• $\lim_{x\to -\infty}F(x)=0$ and $\lim_{x\to \infty}F(x)=1$.
• It is right-continuous.

• For any $a<b$, $$P(a<X\le b)=F(b)-F(a)$$

• For discrete variables, the CDF has jumps at the possible values.

Illustration of a typical CDF shape

F(x)
1.0 |                           ________
    |                        __/
    |                     __/
    |                  __/
    |               __/
    |            __/
    |         __/
0.0 |________/____________________________ x
         values of x increasing

This shape shows how cumulative probability increases as $x$ increases.

2. Probability Density Function

Meaning and interpretation

• A density function $f(x)$ describes how probability is distributed continuously over values.
• Unlike discrete probabilities, $f(x)$ itself is not a probability.

• For a continuous random variable, $$P(X=x)=0$$ for any exact point $x$.

• Probabilities are obtained by integrating the density over intervals.

• Example: If $f(x)$ is large near $x=5$, values near 5 are more likely than values where $f(x)$ is small.

Important properties

• $f(x)\ge 0$ for all $x$.

• The total area under the curve is 1: $$\int_{-\infty}^{\infty} f(x)\,dx = 1$$

• Probability over an interval is the area under the curve: $$P(a\le X\le b)=\int_a^b f(x)\,dx$$

• The CDF and PDF are connected by: $$F(x)=\int_{-\infty}^x f(t)\,dt$$ and, where differentiable, $$f(x)=\frac{d}{dx}F(x)$$

Illustration of a density curve

f(x)
 ^
 |                /\
 |               /  \
 |              /    \
 |             /      \
 |____________/__________\____________> x
              a    b

The shaded area between $a$ and $b$ represents $P(a\le X\le b)$.

3. Relationship Between Distribution Function and Density

How they are connected

• The distribution function is the accumulated area of the density function.
• The density function is the slope of the distribution function wherever the CDF is differentiable.
• This relationship helps move from probabilities to functions and vice versa.
• If a density is known, the CDF can be found by integration.
• If the CDF is known and smooth, the density can be found by differentiation.

Discrete versus continuous cases

• For discrete random variables, the CDF increases in jumps.
• For continuous random variables, the CDF increases smoothly.
• In discrete cases, probabilities are assigned to exact values using a probability mass function rather than a density function.
• In continuous cases, the density function replaces point probabilities with interval probabilities.
• Example:
  • A die roll is discrete.
  • A person’s exact height is modeled as continuous.

Relationship sketch

Density f(x):        CDF F(x):
    /\                   ________
   /  \                __/
  /    \             _/
 /      \          _/
-------------------/------------------> x

The left side shows a density curve; the right side shows the corresponding cumulative increase.

Working / Process

1. Identify the type of random variable

• Determine whether the variable is discrete or continuous.
• If it is discrete, probabilities are found from a probability mass function and the CDF is built from cumulative sums.
• If it is continuous, probabilities are found using a density function and integration.

2. Use the correct function to compute probability

• For a distribution function: $$P(a<X\le b)=F(b)-F(a)$$

• For a density function: $$P(a\le X\le b)=\int_a^b f(x)\,dx$$

• If only the density is given, first find the corresponding cumulative probability by integrating.

3. Check validity and interpret the result

• Ensure the CDF stays between 0 and 1 and is non-decreasing.
• Ensure the PDF is non-negative and has total area 1.
• Interpret the numerical result in context, such as waiting time, height, score, or measurement error.
• Example: If $P(2<X<5)=0.3$, then 30% of outcomes lie between 2 and 5.

Advantages / Applications

They provide a complete description of uncertainty

• Distribution functions summarize the total probability structure of a random variable.
• Densities show where values are concentrated more heavily.

They are widely used in real-world modeling

• Applied in economics, engineering, biology, weather prediction, and quality control.
• Useful for modeling continuous measurements such as time, weight, voltage, and temperature.

They support advanced statistical methods

• Used in expectation, variance, hypothesis testing, confidence intervals, and simulation.
• Help in comparing models and deriving distributions of transformed variables.

Summary

• Distribution functions describe cumulative probability, while densities describe how probability is spread over continuous values.
• A CDF gives $P(X\le x)$, and a PDF gives interval probabilities through area under the curve.
• These functions are closely related by integration and differentiation, making them central to probability and statistics.
• Important terms to remember: random variable, cumulative distribution function, probability density function, interval probability, continuity