Math Problem Statement
Solution
To solve the problem related to the joint probability density function , let us address each part systematically:
(a) Find the value of :
The total probability for a joint probability density function must integrate to 1 over the entire range. Thus, we solve:
Solution:
-
Perform the inner integral with respect to :
-
Perform the outer integral with respect to :
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Set this equal to 1 and solve for :
Thus, .
(b) Find the marginal probability density functions of and :
:
The marginal density is obtained by integrating the joint density over :
- Compute the integral:
Thus, , for .
:
The marginal density is obtained by integrating the joint density over :
- Compute the integral:
Thus, , for .
(c) Find the conditional probability density function :
The conditional density is given by:
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Substitute and :
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Simplify:
Thus, .
Summary of Results:
- (a) .
- (b) Marginal densities:
- .
- .
- (c) Conditional density:
- .
Would you like further explanations or graphical visualizations of these results? Here are five related questions to expand your understanding:
- How would the range of and affect the validity of the joint probability density function?
- Can you verify that the marginal densities and integrate to 1?
- How is the conditional density interpreted in terms of dependence between and ?
- What is the expected value of and using their marginal densities?
- How would you extend this to find or ?
Tip: Always ensure the joint density integrates to 1 across its range before proceeding with marginal or conditional calculations!
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Math Problem Analysis
Mathematical Concepts
Joint Probability Density Functions
Marginal Probability Density
Conditional Probability Density
Formulas
∫∫ f(x, y) dx dy = 1 (Normalization condition for PDF)
f_X(x) = ∫ f(x, y) dy (Marginal PDF of X)
f_Y(y) = ∫ f(x, y) dx (Marginal PDF of Y)
f_{X|Y}(x|y) = f(x, y) / f_Y(y) (Conditional PDF)
Theorems
Law of Total Probability
Normalization Condition of PDF
Suitable Grade Level
Undergraduate
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