1. **Problem Statement:** We are given the cumulative distribution function (CDF) $F(x)$ of a uniform random variable $X \sim \text{Unif}(a,b)$. 2. **Definition of the CDF:** The CDF $F(x)$ is defined as the probability that the random variable $X$ takes a value less than or equal to $x$: $$F(x) = P(X \leq x) = \int_{-\infty}^x f(t) \, dt$$ where $f(t)$ is the probability density function (PDF) of $X$. 3. **PDF of Uniform Distribution:** For $X \sim \text{Unif}(a,b)$, the PDF is: $$f(t) = \begin{cases} \frac{1}{b-a} & \text{if } a < t < b \\ 0 & \text{otherwise} \end{cases}$$ 4. **Evaluating the CDF:** - For $x \leq a$, since $X$ cannot be less than $a$, the CDF is: $$F(x) = 0$$ - For $a < x < b$, integrate the PDF from $a$ to $x$: $$F(x) = \int_a^x \frac{1}{b-a} \, dt = \frac{x - a}{b - a}$$ - For $x \geq b$, the entire probability mass is accumulated, so: $$F(x) = 1$$ 5. **Summary of the CDF:** $$F(x) = \begin{cases} 0 & x \leq a \\ \frac{x - a}{b - a} & a < x < b \\ 1 & x \geq b \end{cases}$$ 6. **Interpretation:** - The CDF is zero before $a$ because the random variable cannot take values less than $a$. - It increases linearly between $a$ and $b$ with slope $\frac{1}{b-a}$ because the PDF is constant. - It reaches 1 at and beyond $b$ because the total probability is 1. This piecewise linear function fully describes the CDF of a uniform distribution on $[a,b]$.