17.4.2. For Continuous Random Variables

Try these first

1. 1.

   What is the definition of independent random variables?

   Hint

   Think about the relationship between their outcomes.

2. 2.

   Give an example of two independent continuous random variables.

   Hint

   Consider measurements that do not interfere with each other.

3. 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)

   Hint

   Remember the formula for independence.

4. 4.

   If f(X,Y) is not equal to f(X) * f(Y), then X and Y are:

   • True
   • False

   Hint

   Think about the condition for independence.

5. 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.

   Hint

   Focus on calculating the marginals carefully.

6. 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.

   Hint

   Reflect on how independence affects the products of expectations.