17.4.2 - For Continuous Random Variables
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Practice Questions
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What is the definition of independent random variables?
💡 Hint: Think about the relationship between their outcomes.
Give an example of two independent continuous random variables.
💡 Hint: Consider measurements that do not interfere with each other.
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Interactive Quizzes
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Two random variables X and Y are independent if:
💡 Hint: Remember the formula for independence.
If f(X,Y) is not equal to f(X) * f(Y), then X and Y are:
💡 Hint: Think about the condition for independence.
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Challenge Problems
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Suppose X and Y have joint PDF given as f(X,Y) = 1/(2πσ^2)e^(-((X-μ_x)^2+(Y-μ_y)^2)/(2σ^2)) for X,Y in ℝ. Prove if X and Y are independent.
💡 Hint: Focus on calculating the marginals carefully.
In a study, varX and varY are found to be independent with PDFs being non-overlapping across all domains. Discuss the implications on their expected values.
💡 Hint: Reflect on how independence affects the products of expectations.
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