Math Problem Statement
Solution
Let's break down and solve each part of this problem.
Problem 1: Joint Probability Density Function
Given:
We are asked to find the expected value .
To find , we use: Substitute :
Steps:
- Expand the integrand .
- Integrate with respect to first, from to .
- Integrate the result with respect to , from to .
Problem 2: Joint Probability Mass Function
The table of the joint probability mass function is given for and .
Part (a): Find
This probability is the sum of values where . From the table:
- For
- For
- For
So, Using the table values:
Part (b): Find the Marginal Probability Density of
The marginal probability of , , is found by summing the probabilities across all values of for each :
- For :
- For :
So, the marginal probability density of is:
Would you like further details on the integration steps for or any specific part?
Related Questions
- What is the marginal probability density of in Problem 2?
- How do we interpret the expected value in a real-world context?
- What conditions make a joint probability density function valid?
- How would and be computed for the distribution in Problem 1?
- In Problem 2, how would we calculate ?
Tip
For joint distributions, remember that marginal probabilities are calculated by summing or integrating over the unwanted variable.
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Math Problem Analysis
Mathematical Concepts
Joint Probability Density Function
Expected Value
Marginal Probability
Formulas
Expected Value E(xy) = ∫∫ xy * f(x, y) dy dx
Marginal Probability p(x) = Σ p(x, y) over y
Theorems
Properties of Joint Probability Distributions
Marginal Probability Calculation
Suitable Grade Level
College Level - Probability and Statistics
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