Math Problem Statement
Answer it sll
Solution
I see that you've uploaded an exercise sheet in French involving probability theory and statistical analysis. Let me walk you through each exercise:
Exercice 1:
Given the random variable pair with a joint density function:
\begin{cases} c, & \text{if } 1 \leq x \leq 2y \leq 1 \\ 0, & \text{otherwise} \end{cases}$$ **1. Trouver la constante $$c$$:** We know the total probability must be 1, so we integrate the joint density function over the domain where it is non-zero and set the integral equal to 1. That means solving for $$c$$ using the following integral: $$\int_{x=1}^{2y} \int_{y=\frac{1}{2}}^{1} c \, dy \, dx = 1$$ You'll need to evaluate the limits of integration. **2. Calculer $$P(X < 0 \, / \, Y < 0.25)$$:** This is asking for the conditional probability $$P(X < 0 \, | \, Y < 0.25)$$, which requires evaluating the joint density over this region and dividing by the marginal probability for $$Y < 0.25$$. **3. Trouver les fonctions de densité marginale:** The marginal density for $$X$$ is found by integrating the joint density function over $$y$$: $$f_X(x) = \int_{y} f_{(X,Y)}(x, y) \, dy$$ Similarly, for $$Y$$, integrate over $$x$$: $$f_Y(y) = \int_{x} f_{(X,Y)}(x, y) \, dx$$ You'll also need to justify whether $$X$$ and $$Y$$ are independent by checking if the joint density factorizes into the product of the marginals. **4. Donner le modèle de régression de $$Y$$ sur $$X$$:** This involves determining the regression model for $$Y$$ given $$X$$. This can be done by calculating $$E(Y|X)$$. --- ### **Exercice 2:** The joint distribution of $$(X,Y)$$ is given in a table: | | -1 | 0 | 1 | Total | |---|---|---|---|---| |-1 | a | 2a | 4a | 4a | |1 | 3a | 0 | 4a | 4a | |Total | ... | ... | ... | 4a | **1. Reportez le tableau et déterminez $$a$$:** The total probability for each row must sum to 1, so by solving the row totals, you can determine the value of $$a$$. **2. Calculer $$E(X)$$, $$E(Y)$$:** The expected value of $$X$$ is given by: $$E(X) = \sum_{x,y} x \cdot P(X = x, Y = y)$$ Similarly for $$Y$$. **3. Étudier l’indépendance de $$X$$ et $$Y$$:** To check if $$X$$ and $$Y$$ are independent, compare the joint probability $$P(X, Y)$$ with the product of the marginals $$P(X)$$ and $$P(Y)$$. --- ### **Exercice 3:** Analyze the numerical and graphical statistics for the given dataset: $$2, 4, 4, 6, 7, 10, 11, 12$$ You are likely asked to calculate summary statistics such as mean, variance, and construct graphs (histogram, box plot, etc.). --- Would you like me to guide you through the detailed steps for any of the exercises?Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Probability Density Functions
Conditional Probability
Marginal Distributions
Expected Values
Statistical Regression
Discrete Probability Distributions
Formulas
Total Probability Integral: ∫∫ f(x, y) dx dy = 1
Conditional Probability: P(A | B) = P(A ∩ B) / P(B)
Expected Value: E(X) = Σ x * P(X = x)
Marginal Distribution Formula: f_X(x) = ∫ f(x, y) dy
Theorems
Independence of Random Variables
Law of Total Probability
Suitable Grade Level
University (Undergraduate Level)
Related Recommendation
Joint Density Function and Conditional Expectation of X and Y
Solving Probability Theory Problems: CDF, Expected Value, and Markov's Inequality
Variance, Covariance, and Probability Analysis of X and Y Random Variables
Calculating Expected Values, Covariance, and Independence of Random Variables
Joint Probability Distribution: Marginal Distributions, Expectation, Covariance, and Correlation of X and Y