Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

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 \([a, b]\).

Step-by-Step Solution

Answer

The median is \( x_0 = \frac{a+b}{2} \).

Step 1: Understanding Uniform Distribution

A uniform distribution on the interval \([a, b]\) is a distribution where all outcomes are equally likely. The probability density function (PDF) of a uniform distribution is constant and given by \[ f(x) = \frac{1}{b-a} \] for \( a \leq x \leq b \).

Step 2: Understanding the Median of a Distribution

The median of a continuous random variable \(X\) is the value \(x_0\) such that the cumulative distribution function (CDF) equals 0.5. For a uniform distribution, this means finding \(x_0\) such that \(P(X \leq x_0) = 0.5\).

Step 3: Calculate the Median Using the Cumulative Distribution Function

The CDF for a uniform distribution between \(a\) and \(b\) is given by \[ F(x) = \frac{x-a}{b-a} \]for \( a \leq x \leq b \). We need to solve for \(x_0\) in the equation \[ \frac{x_0-a}{b-a} = 0.5 \].

Step 4: Solve for the Median

Solving the equation \( \frac{x_0 - a}{b-a} = 0.5 \), we multiply both sides by \((b-a)\) to isolate \(x_0\). This gives us: \[ x_0 - a = 0.5(b-a) \].

Step 5: Simplify the Expression

Add \(a\) to both sides of the equation to get \[ x_0 = 0.5(b-a) + a \].

Step 6: Final Expression of the Median

The expression simplifies to \[ x_0 = \frac{a+b}{2} \]. Therefore, the median of a uniform distribution on the interval \([a, b]\) is the midpoint of the interval.

Key Concepts

Continuous Random Variable

A continuous random variable is a type of variable that can take an infinite number of possible values within a given range. Imagine plotting points on a line; a continuous random variable represents any value along that line instead of at distinct intervals. For example, the temperature in a city is continuous because it can be 30.1°C, 30.15°C, 30.155°C, and so on without limitation. This is unlike a discrete random variable, such as the number of students in a classroom, where values are distinct and separate.

• Characteristics: Can take any real value.
• Measured: Not counted.
• Examples: Heights, weights, incomes.

For continuous random variables, the entire range between minimum and maximum values must be considered. This is where distributions come into play, with the normal and uniform distributions being common examples.

Probability Density Function

The probability density function (PDF) is a fundamental concept in understanding continuous random variables. It describes the likelihood of a continuous random variable taking on a specific value. In simpler terms, it's like a roadmap showing us where we are more likely to "land" on a continuous line of possibilities.

• Function Form: For a uniform distribution on \[a, b\], the PDF is \ \[ f(x) = \frac{1}{b-a} \ \] if \ \ a \leq x \leq b \.
• Properties: The area under the PDF curve over its entire range equals 1.
• Application: Used to calculate probabilities for continuous data within specific intervals.

In the case of a uniform distribution, the PDF is flat, signifying equal probability for all outcomes within the defined range \[a, b\]. This makes computation straightforward since each outcome has an identical likelihood.

Cumulative Distribution Function

The cumulative distribution function (CDF) is another essential tool when dealing with continuous random variables. If the PDF tells us where we're likely to land, the CDF tells us how probable it is to land anywhere from the minimum value up to a certain point. It's like accumulating probabilities as you move along the number line.

• Function form: For a uniform distribution on \[a, b\], the CDF is given by \ \[ F(x) = \frac{x-a}{b-a} \ \] for \ \ a \leq x \leq b \.
• Properties:
  • The CDF always ranges from 0 to 1.
  • As \ \ x \ \ increases, \ \ F(x) \ \ also increases.
• Usage: Helps in determining percentile ranks and probabilities below a certain value.

For a uniform distribution, as you move from \ \ a \ \ to \ \ b \, the CDF steadily rises from 0 to 1, showing how the likelihood accumulates over the interval as expected.

Median of Distribution

The median of a distribution is a measure of central tendency indicating the midpoint of the data distribution, separating it into two equal halves. For continuous random variables, finding the median means finding the value where exactly half the data lies below it and half above it.

• Definition: For a continuous random variable \ \ X \ \, the median \ \ x_0 \ \ satisfies\ \[ P\left(X \leq x_0\right) = 0.5 \ \].
• Importance: Provides a robust central value that is not affected by outliers.
• Uniform Distribution Example: For a distribution on the interval \ \ [a, b] \, the median is \ \ x_0 = \frac{a+b}{2} \ , being exactly at the midpoint.

In a uniform distribution, the median is straightforward. It's simply the average of the interval boundaries \ \ a \ \ and \ \ b \, dividing the distribution equally in terms of area, or probability, on either side. This balance makes it easy to locate the median in uniform distributions.