2023-11-15
Show that a joint pdf consists of two independent, continuous RVs.
Combine two independent RVs into one joint pdf or CDF.
What do we know about independence for events and discrete RVs?
For events: If \(A \perp B\)
\[P(A \cap B) = P(A)P(B)\] \[P(A|B) = P(A)\]
For discrete RVs: If \(X \perp Y\) \[p_{X,Y}(x,y) = p_{X}(x)p_{Y}(y)\] \[F_{X,Y}(x,y) = F_{X}(x)F_{Y}(y)\] \[p_{X|Y}(x|y) = p_{X}(x)\] \[p_{Y|X}(y|x) = p_{Y}(y)\]
What does it mean for continuous r.v.’s to be independent?
For continuous RVs: If \(X \perp Y\)
Example 1.1
Let \(X\) and \(Y\) be independent r.v.’s with \(f_X(x)= \frac12\), for \(0 \leq x \leq 2\) and \(f_Y(y)= 3y^2\), for \(0 \leq y \leq 1\).
Example 1.2
Let \(X\) and \(Y\) be independent r.v.’s with \(f_X(x)= \frac12\), for \(0 \leq x \leq 2\) and \(f_Y(y)= 3y^2\), for \(0 \leq y \leq 1\).
Example 2.1
Let \(f_{X,Y}(x,y)= 18 x^2 y^5\), for \(0 \leq x \leq 1, \ 0 \leq y \leq 1\).
Example 2.2
Let \(f_{X,Y}(x,y)= 18 x^2 y^5\), for \(0 \leq x \leq 1, \ 0 \leq y \leq 1\).
Do this problem at home for extra practice. The solution is available in Meike’s video!
Example 3
Let \(f_{X,Y}(x,y)= 2 e^{-(x+y)}\), for \(0 \leq x \leq y\). Are \(X\) and \(Y\) independent?
If \(f_{X,Y}(x,y)= g(x)h(y)\), where \(g(x)\) and \(h(y)\) are pdf’s, then \(X\) and \(Y\) are independent.
If \(F_{X,Y}(x,y)= G(x)H(y)\), where \(G(x)\) and \(H(y)\) are cdf’s, then \(X\) and \(Y\) are independent.
Chapter 26 Slides