Math Problem Statement
Soit \( \alpha > 2 \). On considère la fonction \( f \) définie par:
\[
f(x) =
\begin{cases}
\frac{\alpha - 1}{x^\alpha}, & x \geq 1 \\
0, & x < 1
\end{cases}
\]
1. Montrer que \( f \) est une densité de probabilité d'une variable \( X \).
2. Déterminer la fonction de répartition \( F_X \).
3. Calculer l'espérance de \( X \).
Solution
This problem involves a piecewise probability density function (PDF) defined as:
\begin{cases} \frac{\alpha - 1}{x^\alpha}, & x \geq 1 \\ 0, & x < 1 \end{cases}$$ where $$\alpha > 2$$. ### Task 1: Show that $$f(x)$$ is a valid probability density function. A function is a PDF if: 1. $$f(x) \geq 0$$ for all $$x$$, and 2. The integral of $$f(x)$$ over all $$x$$ equals 1: $$\int_{-\infty}^\infty f(x) \, dx = 1.$$ Since $$f(x) = 0$$ for $$x < 1$$, we only need to check $$x \geq 1$$: $$\int_1^\infty \frac{\alpha - 1}{x^\alpha} dx.$$ Let’s compute this integral: $$\int_1^\infty \frac{\alpha - 1}{x^\alpha} dx = (\alpha - 1) \int_1^\infty x^{-\alpha} dx.$$ For $$\alpha > 1$$, the integral $$\int x^{-\alpha} dx$$ converges: $$\int x^{-\alpha} dx = \frac{x^{1-\alpha}}{1-\alpha}.$$ Evaluate at the bounds $$1$$ to $$\infty$$: $$\int_1^\infty x^{-\alpha} dx = \left[ \frac{x^{1-\alpha}}{1-\alpha} \right]_1^\infty = \frac{1}{\alpha - 1}.$$ Thus: $$\int_1^\infty f(x) dx = (\alpha - 1) \cdot \frac{1}{\alpha - 1} = 1.$$ Hence, $$f(x)$$ is a valid PDF. --- ### Task 2: Determine the cumulative distribution function $$F_X(x)$$. The cumulative distribution function $$F_X(x)$$ is defined as: $$F_X(x) = \int_{-\infty}^x f(t) dt.$$ For $$x < 1$$: $$F_X(x) = 0 \quad \text{(since $$f(t) = 0$$ for $$t < 1$$)}.$$ For $$x \geq 1$$: $$F_X(x) = \int_1^x \frac{\alpha - 1}{t^\alpha} dt.$$ Compute the integral: $$\int_1^x t^{-\alpha} dt = \left[ \frac{t^{1-\alpha}}{1-\alpha} \right]_1^x = \frac{x^{1-\alpha} - 1}{1-\alpha}.$$ Thus: $$F_X(x) = (\alpha - 1) \cdot \frac{x^{1-\alpha} - 1}{1-\alpha} = 1 - x^{1-\alpha}, \quad x \geq 1.$$ So: $$F_X(x) = \begin{cases} 0, & x < 1, \\ 1 - x^{1-\alpha}, & x \geq 1. \end{cases}$$ --- ### Task 3: Calculate the expectation $$\mathbb{E}[X]$$. The expectation is given by: $$\mathbb{E}[X] = \int_{-\infty}^\infty x f(x) dx.$$ Since $$f(x) = 0$$ for $$x < 1$$, we only integrate over $$x \geq 1$$: $$\mathbb{E}[X] = \int_1^\infty x \cdot \frac{\alpha - 1}{x^\alpha} dx = (\alpha - 1) \int_1^\infty x^{1-\alpha} dx.$$ Using the earlier result for $$\int x^{-\alpha} dx$$: $$\int_1^\infty x^{1-\alpha} dx = \frac{x^{2-\alpha}}{2-\alpha} \Big|_1^\infty = \frac{1}{\alpha - 2}.$$ Thus: $$\mathbb{E}[X] = (\alpha - 1) \cdot \frac{1}{\alpha - 2} = \frac{\alpha - 1}{\alpha - 2}.$$ --- ### Final Results: 1. $$f(x)$$ is a valid probability density function. 2. The cumulative distribution function $$F_X(x)$$ is: $$F_X(x) = \begin{cases} 0, & x < 1, \\ 1 - x^{1-\alpha}, & x \geq 1. \end{cases}$$ 3. The expectation $$\mathbb{E}[X]$$ is: $$\mathbb{E}[X] = \frac{\alpha - 1}{\alpha - 2}.$$ --- Do you have any questions or want detailed steps for a specific part? ### Related Questions: 1. What is the variance $$\text{Var}(X)$$ of this distribution? 2. How does $$F_X(x)$$ change as $$\alpha \to \infty$$? 3. What happens to $$\mathbb{E}[X]$$ if $$\alpha$$ approaches its lower bound (e.g., $$\alpha \to 2^+$$)? 4. Can we derive higher moments of $$X$$, like $$\mathbb{E}[X^2]$$? 5. How does the behavior of $$f(x)$$ compare to a uniform distribution for large $$\alpha$$? ### Tip: When verifying if a function is a PDF, always confirm the normalization condition using proper integration techniques.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Probability Theory
Integration
Cumulative Distribution Functions
Expected Value
Formulas
Normalization condition for PDF: \( \int_{-\infty}^{\infty} f(x) \, dx = 1 \)
CDF: \( F_X(x) = \int_{-\infty}^x f(t) \, dt \)
Expected value: \( \mathbb{E}[X] = \int_{-\infty}^{\infty} x f(x) \, dx \)
Theorems
Properties of Probability Density Functions
Fundamental Theorem of Calculus
Suitable Grade Level
University Level
Related Recommendation
Test Whether a Function Can Be a Probability Density Function
Probability Density Function and Cumulative Distribution Function Calculation
Finding Probability Density, Expectation, and Variance from CDF of Random Variable X
Probability Distribution: Finding Parameters and Key Characteristics
Finding Constant k and Cumulative Distribution Function (CDF) for Continuous Random Variable X