Suppose the probability density function is $f\left(x\right)=\frac{1}{2}$ for $0<x<2$. Which of the following statements is true about the median of $x$?

• A. The median of $x$ is $2$.
• B. The median of $x$ is $0$.
• C. The median of $x$ is $1$.
• D. The median of $x$ is $0.5$.

Answer: (C)

Verified step by step guidance

1. Identify the given probability density function (pdf): \(f(x) = \frac{1}{2}\) for \$0 < x < 2$. This means the distribution is uniform over the interval (0, 2).
2. Recall that the median \(m\) of a continuous random variable \(X\) is the value such that the cumulative distribution function (CDF) satisfies \(P(X \leq m) = 0.5\).
3. Find the cumulative distribution function (CDF) by integrating the pdf from the lower bound up to \(x\): \(F(x) = \int_0^x \frac{1}{2} \, dt = \frac{x}{2}\) for \$0 < x < 2$.
4. Set the CDF equal to 0.5 to find the median: \(F(m) = \frac{m}{2} = 0.5\).
5. Solve the equation \(\frac{m}{2} = 0.5\) for \(m\) to find the median value.