CDF of Continuous Uniform Random Variable [a,b]

Math Problem Statement

Let X be the continuous uniform random variable on [a,b]. The cumulative distribution function (CDF) of X is?

The question asks for the cumulative distribution function (CDF) of a continuous uniform random variable $X$ over the interval $[a, b]$ .

The probability density function (PDF) of $X$ is:

\frac{1}{b-a}, & a \leq x \leq b, \\ 0, & \text{otherwise}. \end{cases}$$ The cumulative distribution function (CDF) is found by integrating the PDF: $$F(x) = P(X \leq x) = \int_a^x f(t) \, dt = \int_a^x \frac{1}{b-a} \, dt.$$ ### Calculation of $$F(x)$$: 1. **Integrate $$f(t) = \frac{1}{b-a}$$ from $$a$$ to $$x$$:** $$F(x) = \frac{1}{b-a} \int_a^x dt = \frac{1}{b-a} \big[x - a\big].$$ 2. **Result:** $$F(x) = \frac{x - a}{b - a}, \quad \text{for } x \in [a, b].$$ ### Answer: The correct option is: $$\textbf{a. } \frac{x-a}{b-a}.$$ Let me know if you'd like further explanation or have other questions! --- Here are five related questions for practice: 1. How do you derive the mean and variance of a uniform distribution? 2. What is the CDF of $$X$$ if $$X$$ is uniform on $$[0, 1]$$? 3. How would the PDF and CDF change if the interval of $$X$$ is $$[-1, 1]$$? 4. What is the inverse CDF (quantile function) for a uniform distribution? 5. Can the CDF ever decrease for a valid random variable? **Tip:** The CDF is always a non-decreasing function because probabilities cannot decrease.

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Continuous uniform distribution
Cumulative distribution function (CDF)

F(x) = (x - a) / (b - a) for x in [a, b]

Definition of cumulative distribution function (CDF)
Properties of uniform distributions

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