14.3.2. Marginal PDF (Continuous)

Try these first

1. 1.

   What is the marginal PDF of X if the joint PDF is f(x,y) = 6xy for 0 ≤ x, y ≤ 1?

   Hint

   Integrate 6xy with respect to y from 0 to 1.

2. 2.

   State the condition for marginal PDFs in relation to joint PDFs.

   Hint

   Consider the definition of probability density functions.

3. 3.

   What does the marginal PDF represent?

   • The probability distribution of multiple variables
   • The distribution of a single variable from a joint distribution
   • The cumulative distribution function

   Hint

   Think about the term 'marginal' and its meaning in statistics.

4. 4.

   Integration is used to find marginal PDFs because it allows for:

   • True
   • False

   Hint

   Recall the purpose of integration in mathematics.

5. 5.

   Given the joint probability density function f(x,y) = 3x^2y for 0<x<1 and 0<y<1, derive f_X(x) and f_Y(y).

   Hint

   Check your limits and ensure proper integration across defined boundaries.

6. 6.

   A researcher calculates the marginal PDF of X as f_X(x)=x^2 for x≥0. If the joint PDF integrates to 1, what does it indicate about Y?

   Hint

   Remember that joint distributions always require summation to 1 across all dimensions.