Consider a joint probability density function (PDF) for continuous random variables  X
and  Y
, given by

fX,Y(x,y)=c(1−x2)(1−y2)

for  0≤x≤1
and  0≤y≤1
, and  fX,Y(x,y)=0
otherwise, where  c
is constant.

(a) Determine the value of  c
to ensure that  fX,Y
is a valid joint probability density function.

c =

To determine the value of \( c \) such that \( f_{X,Y}(x,y) = c(1 - x^2)(1 - y^2) \) is a valid joint probability density function, we need to ensure that the total probability integrates to 1 over the entire support of the random variables \( X \) and \( Y \).

Given the support is \( 0 \leq x \leq 1 \) and \( 0 \leq y \leq 1 \), we need to solve the following integral:

\[
\int_{0}^{1} \int_{0}^{1} f_{X,Y}(x,y) \, dx \, dy = 1
\]

Substituting the expression for \( f_{X,Y}(x,y) \):

\[
\int_{0}^{1} \int_{0}^{1} c(1 - x^2)(1 - y^2) \, dx \, dy = 1
\]

This can be separated into two integrals:

\[
c \int_{0}^{1} (1 - x^2) \, dx \int_{0}^{1} (1 - y^2) \, dy = 1
\]

First, evaluate the integral with respect to \( x \):

\[
\int_{0}^{1} (1 - x^2) \, dx = \left[ x - \frac{x^3}{3} \right]_{0}^{1} = \left( 1 - \frac{1}{3} \right) - (0 - 0) = \frac{2}{3}
\]

Next, evaluate the integral with respect to \( y \):

\[
\int_{0}^{1} (1 - y^2) \, dy = \left[ y - \frac{y^3}{3} \right]_{0}^{1} = \left( 1 - \frac{1}{3} \right) - (0 - 0) = \frac{2}{3}
\]

Now multiply these results together:

\[
c \left( \frac{2}{3} \right) \left( \frac{2}{3} \right) = 1
\]

\[
c \cdot \frac{4}{9} = 1
\]

Solving for \( c \):

\[
c = \frac{9}{4}
\]

Thus, the value of \( c \) that ensures \( f_{X,Y}(x,y) \) is a valid joint probability density function is:

\[
\boxed{c = \frac{9}{4}}
\]

Would you like more details or have any questions? 

Here are 5 related questions:
1. How do you find the marginal PDF from a joint PDF?
2. What is the interpretation of a joint probability density function?
3. How would you determine the expected value of \( X \) using this joint PDF?
4. What conditions must a function satisfy to be a valid joint PDF?
5. How can you calculate the correlation coefficient between \( X \) and \( Y \) given their joint PDF?

**Tip:** Always check the limits of integration when working with joint PDFs to ensure they cover the entire support of the random variables.

Learn how to determine the constant c for the joint probability density function fX,Y(x,y) = c(1 - x^2)(1 - y^2), ensuring it integrates to 1 over the support [0,1]x[0,1]. This problem involves advanced concepts in probability theory and requires integration techniques.

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