Question

The median of a continuous random variable \(X\) is a value \(x_{0}\) such that \(P\left(X \leq x_{0}\right)=0.5 .\) Find the median of a uniform random variable on the interval \([a, b]\).

Short answer

The median of a uniform random variable on \([a, b]\) is \(\frac{a+b}{2}\).

Step-by-step solution

Understanding the Uniform Distribution

A uniform random variable on the interval \([a, b]\) has a probability density function (PDF) that is constant over the interval. The PDF is given by \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\), and zero otherwise. This means that any subinterval of \([a, b]\) has equal probability density.

Defining the Median Condition

The median \(x_0\) is defined such that \(P(X \leq x_0) = 0.5\). This means that exactly half of the probability is to the left of \(x_0\), and half is to the right in the interval \([a, b]\).

Finding the Cumulative Distribution Function

The cumulative distribution function (CDF) for a uniform distribution from \(a\) to \(b\) is \(F(x) = \frac{x-a}{b-a}\) for \(a \leq x \leq b\). This function gives the probability that the random variable \(X\) is less than or equal to \(x\).

Setting Up the Equation for the Median

We need to solve the equation \(F(x_0) = 0.5\). Substituting the CDF, this becomes \(\frac{x_0 - a}{b-a} = 0.5\).

Solving for the Median

Rearranging the equation, we get: \[ x_0 - a = 0.5(b-a) \] Solving for \(x_0\), we add \(a\) to both sides: \[ x_0 = 0.5(b-a) + a \] This simplifies to \(x_0 = \frac{a+b}{2}\), which is exactly the midpoint of the interval.

Key concepts

Understanding the Median of a Uniform Distribution

In statistics and probability, the median is a vital measure to understand. For a continuous random variable, the median is the point at which the probability of the variable being less than or equal to it is 0.5. This means half of the possible outcomes lie below this value, and half lie above it.
In the context of a uniform distribution defined over the interval \[a, b\], the median is particularly straightforward to determine. Since this distribution is constant across its range, the median simplifies to the midpoint of the interval. By solving the median condition, where \((X \leq x_0) = 0.5\), we find that the median \(x_0\) is given by the formula \(\frac{a+b}{2}\).
Hence, understanding the median in a uniform distribution helps us appreciate the symmetrical nature of this distribution, making it a key characteristic in probability analysis.

Cumulative Distribution Function (CDF) Fundamentals

The Cumulative Distribution Function, or CDF, is an essential concept when discussing probability distributions. It describes the probability that a random variable takes on a value less than or equal to a certain point. For a uniform distribution over the interval \[a, b\], the CDF is expressed as \(F(x) = \frac{x-a}{b-a}\) where \(a \leq x \leq b\).
This function increases linearly from 0 to 1 as \(x\) moves from \(a\) to \(b\), reflecting the uniform probability spread. The CDF is useful because it provides a complete overview of the probability distribution, allowing us to analyze the likelihood of different outcomes. By setting the CDF equal to 0.5, we can locate the median, as discussed earlier.
The CDF thus acts as a powerful tool for understanding and calculating various statistics related to probability distributions.

Probability Density Function (PDF) Explained

The Probability Density Function, or PDF, is another cornerstone concept in understanding probability distributions, particularly for continuous random variables. For a uniform distribution over the interval \[a, b\], the PDF is defined as a constant function, \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\), and zero otherwise.
This constant value indicates that every outcome within the interval \[a, b\] has the same likelihood of occurring, which is why it's called a 'uniform' distribution. The PDF can be thought of as describing how the probability is distributed across the range of a random variable.
While the PDF provides the likelihood of a specific outcome, when integrated over a range, it relates to the CDF which gives the cumulative probability. Understanding the PDF's role is crucial for calculating probabilities and comprehending the nature of the distribution itself.