In the realm of probability and statistics, the uniform distribution stands out as one of the simplest yet most fundamental distributions. It describes a scenario where all outcomes within a specific range are equally likely. To fully grasp this distribution, understanding its Cumulative Distribution Function (CDF) is crucial. This article delves into the CDF of a continuous uniform distribution, providing a clear explanation and proof.
What is a Uniform Distribution?
A continuous uniform distribution, denoted as $mathcal{U}(a, b)$, is characterized by two parameters: $a$ (the minimum value) and $b$ (the maximum value), where $a < b$. Imagine picking a number at random from the interval $[a, b]$ such that every number in this interval has the same chance of being selected. This is precisely what a uniform distribution models.
The probability density function (PDF) of a continuous uniform distribution is constant within the interval $[a, b]$ and zero elsewhere. Mathematically, it’s defined as:
[
mathcal{U}(x; a, b) = left{
begin{array}{rl}
frac{1}{b-a} ; , & text{if} ; a leq x leq b
0 ; , & text{otherwise} ; .
end{array}
right.
]
The Cumulative Distribution Function (CDF) Defined
The Cumulative Distribution Function (CDF), often denoted as $F_X(x)$, for a random variable $X$, gives the probability that $X$ will take a value less than or equal to $x$. In simpler terms, it accumulates the probabilities up to a certain point $x$. For a continuous distribution, the CDF is calculated as the integral of the PDF from $-infty$ to $x$:
[
FX(x) = P(X leq x) = int{-infty}^{x} f_X(z) , mathrm{d}z
]
where $f_X(z)$ is the probability density function of $X$.
CDF of a Continuous Uniform Distribution: Theorem and Proof
For a random variable $X$ following a continuous uniform distribution $mathcal{U}(a, b)$, the CDF is given by:
[
F_X(x) = left{
begin{array}{rl}
0 ; , & text{if} ; x < a
frac{x-a}{b-a} ; , & text{if} ; a leq x leq b
1 ; , & text{if} ; x > b ; .
end{array}
right.
]
Let’s break down the proof step-by-step to understand how we arrive at this formula.
Proof:
We need to consider three cases for the value of $x$:
Case 1: $x < a$
If $x$ is less than the lower bound $a$ of the uniform distribution, then there is no probability for $X$ to be less than or equal to $x$, because the uniform distribution is zero for values less than $a$. Therefore, the CDF in this case is:
[
FX(x) = int{-infty}^{x} mathcal{U}(z; a, b) , mathrm{d}z = int_{-infty}^{x} 0 , mathrm{d}z = 0
]
Case 2: $a leq x leq b$
When $x$ falls within the interval $[a, b]$, the CDF is calculated by integrating the PDF from $-infty$ to $x$. We can split the integral into two parts: from $-infty$ to $a$ and from $a$ to $x$.
[
FX(x) = int{-infty}^{x} mathcal{U}(z; a, b) , mathrm{d}z = int{-infty}^{a} mathcal{U}(z; a, b) , mathrm{d}z + int{a}^{x} mathcal{U}(z; a, b) , mathrm{d}z
]
Since $mathcal{U}(z; a, b) = 0$ for $z < a$, the first integral becomes zero. For $a leq z leq b$, $mathcal{U}(z; a, b) = frac{1}{b-a}$. Thus,
[
begin{split}
FX(x) &= int{-infty}^{a} 0 , mathrm{d}z + int{a}^{x} frac{1}{b-a} , mathrm{d}z
&= 0 + frac{1}{b-a} int{a}^{x} , mathrm{d}z
&= frac{1}{b-a} [z]_a^x
&= frac{1}{b-a} (x – a)
&= frac{x-a}{b-a}
end{split}
]
Case 3: $x > b$
If $x$ is greater than the upper bound $b$, it means we are considering the probability of $X$ being less than or equal to a value that is beyond the entire range of the uniform distribution. In this case, the CDF accumulates the probability over the entire range $[a, b]$, which must equal 1 (certainty). We can see this mathematically as:
[
begin{split}
FX(x) &= int{-infty}^{x} mathcal{U}(z; a, b) , mathrm{d}z = int{-infty}^{b} mathcal{U}(z; a, b) , mathrm{d}z + int{b}^{x} mathcal{U}(z; a, b) , mathrm{d}z
&= FX(b) + int{b}^{x} 0 , mathrm{d}z
&= frac{b-a}{b-a} + 0
&= 1
end{split}
]
This completes the proof for all three cases, confirming the piecewise definition of the CDF for a continuous uniform distribution.
Understanding the CDF Formula
The Cdf Of A Uniform Distribution offers valuable insights:
- For any value $x$ less than $a$, the probability of $X$ being less than or equal to $x$ is 0.
- As $x$ increases from $a$ to $b$, the CDF increases linearly from 0 to 1. The rate of increase is determined by $frac{1}{b-a}$, which is related to the width of the uniform distribution.
- For any value $x$ greater than $b$, the probability of $X$ being less than or equal to $x$ is 1, signifying certainty.
In conclusion, the CDF of a uniform distribution provides a comprehensive view of the probabilities associated with this fundamental distribution. Understanding this function is essential for various statistical applications and analyses involving uniformly distributed random variables.
