Math Problem Statement
The formula for the probability density function for the uniform distribution is:
The formula for the probability density function for the uniform distribution is:
for a ≤ x ≤ b
for a ≤ x ≤ b
for a ≤ x ≤ b
for a ≤ x ≤ b
Solution
The formula for the probability density function (PDF) of a uniform distribution on the interval is given by:
\begin{cases} \frac{1}{b - a} & \text{for } a \leq x \leq b, \\ 0 & \text{otherwise}. \end{cases}$$ ### Explanation: - **Uniform distribution** means every outcome between $$a$$ and $$b$$ is equally likely. - The probability density function (PDF) is constant between $$a$$ and $$b$$ since all intervals have equal likelihood. - The total area under the curve of the PDF must be 1 (since probabilities sum to 1), which explains the factor $$\frac{1}{b - a}$$. ### Key Points: 1. For $$x$$ between $$a$$ and $$b$$, the PDF is a constant value of $$\frac{1}{b - a}$$. 2. Outside of the interval $$[a, b]$$, the PDF is 0. 3. The expected value (mean) of a uniform distribution is $$\frac{a + b}{2}$$. 4. The variance of a uniform distribution is $$\frac{(b - a)^2}{12}$$. 5. This distribution is often used in situations where outcomes are evenly spread across a given range. --- Would you like more details, or do you have any specific questions about the uniform distribution? Here are 5 related questions to consider: 1. How would you compute the cumulative distribution function (CDF) of a uniform distribution? 2. What is the expected value for a uniform distribution, and how is it derived? 3. How does the uniform distribution compare to a normal distribution in terms of shape? 4. What real-world examples can be modeled using a uniform distribution? 5. How do we calculate the variance for a uniform distribution? **Tip**: When working with PDFs, always remember that the total probability (area under the curve) must equal 1 over the range of possible values.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Probability Density Functions
Uniform Distribution
Formulas
f(x) = 1 / (b - a) for a ≤ x ≤ b
Expected Value: E(X) = (a + b) / 2
Variance: Var(X) = (b - a)^2 / 12
Theorems
Uniform Distribution Theorem
Properties of Probability Density Functions
Suitable Grade Level
Undergraduate Level
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