Practice For Continuous Random Variables (17.4.2) - Independence of Random Variables

Practice: For Continuous Random Variables

Lesson 17.4.2Independence of Random Variables

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8 questions

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6 sample questions

Question 1.
What is the definition of independent random variables?
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Think about the relationship between their outcomes.
Question 2.
Give an example of two independent continuous random variables.
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Consider measurements that do not interfere with each other.
3Question 3.
Two random variables X and Y are independent if:
- f(X,Y) = f(X) + f(Y)
- f(X,Y) = f(X) * f(Y)
- f(X,Y) = f(X) - f(Y)
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Remember the formula for independence.
4Question 4.
If f(X,Y) is not equal to f(X) * f(Y), then X and Y are:
- True
- False
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Think about the condition for independence.
Question 5.
Suppose X and Y have joint PDF given as f(X,Y) = 1/(2π*σ^2)*e^(-((X-μ_x)^2+(Y-μ_y)^2)/(2σ^2)) for X,Y in ℝ. Prove if X and Y are independent.
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Focus on calculating the marginals carefully.
Question 6.
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.
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Reflect on how independence affects the products of expectations.

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