Math Problem Statement

The continuous random variable 𝑋𝑋 has probability density function 𝑓𝑓 given by 𝑓𝑓(π‘₯π‘₯) = οΏ½ |π‘Žπ‘Žπ‘Žπ‘Ž| βˆ’3 β©½ π‘₯π‘₯ β©½ 3 0 otherwise where π‘Žπ‘Ž is a constant. The value of π‘Žπ‘Ž is:

Solution

To find the value of aa, we use the fact that the total probability for any probability density function (PDF) must be 1, i.e.,

βˆ«βˆ’βˆžβˆžf(x) dx=1\int_{-\infty}^{\infty} f(x) \, dx = 1

Given the probability density function f(x)f(x):

\begin{cases} a & -3 \leq x \leq 3 \\ 0 & \text{otherwise} \end{cases}$$ We focus on the range where $$f(x) = a$$ (i.e., between -3 and 3), and integrate over this range: $$\int_{-3}^{3} a \, dx = 1$$ Since $$a$$ is constant, we can factor it out of the integral: $$a \int_{-3}^{3} 1 \, dx = 1$$ The integral of 1 from -3 to 3 is simply the length of the interval: $$a \cdot (3 - (-3)) = 1$$ $$a \cdot 6 = 1$$ Solving for $$a$$: $$a = \frac{1}{6}$$ Thus, the value of $$a$$ is $$\frac{1}{6}$$. --- Would you like further details or have any questions? Here are 5 related questions you could explore: 1. How would you compute the expected value $$E(X)$$ for this PDF? 2. What is the variance $$\text{Var}(X)$$ for this distribution? 3. How would you calculate the median of the distribution $$X$$? 4. How does the value of $$a$$ affect the shape of the distribution? 5. Can you extend this to find the cumulative distribution function (CDF)? **Tip**: When working with probability density functions, always remember that the integral of the PDF over the entire possible range of $$X$$ must equal 1!

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Math Problem Analysis

Mathematical Concepts

Probability
Probability Density Function
Integration

Formulas

∫[βˆ’βˆž,∞] f(x) dx = 1
∫[βˆ’3,3] a dx = 1

Theorems

Normalization of Probability Density Function

Suitable Grade Level

Undergraduate - Statistics

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