2.1 Marginal and Conditional Distribution Functions
Consider the partitioned vector \(\underline {X}'=\begin {pmatrix} \underline {X}_1, & \underline {X}_2\\ \end {pmatrix}\) where \(\underline {X}_1\) and \(\underline {X}_2\) have \(k\) and \(p-k\) elements, respectively \((k<p)\) . Let \(\underline {X}\) have a joint pdf \(f\big (\underline {X}\big )\). Then marginal pdf of \(\underline {X}_1\) is given by \begin {equation} \tag {4} f_1\big (\underline {X}_1\big ) = \int ^{\infty }_{-\infty } f\big (\underline {X}_1,\underline {X}_2\big )d\underline {X}_2 \end {equation} the marginal pdf of \(\underline {X}_2\) is defined similarly as
\[f_2\big (\underline {X}_2\big ) = \int ^{\infty }_{-\infty } f\big (\underline {X}_1,\underline {X}_2\big )d\underline {X}_1\]
\(\bullet \) For the given value of \(\underline {X}_1\) , say \(\underline {X}_1 = \underline {X}^0_1\). The conditional pdf of \(\underline {X}_2\) given \(\underline {X}_1 = \underline {X}_1^0\) is \begin {equation} \tag {5} f\big (\underline {X}_2/\underline {X}_1=\underline {X}_1^0\big ) = \frac {f\big (\underline {X}_1^0,\underline {X}_2\big )}{f_1\big (\underline {X}_1^0\big )} \end {equation} where \(f_1\big (\underline {X}_1^0\big ) \neq 0\) .
The conditional pdf of \(\underline {X}_1\) given \(\underline {X}_2 = \underline {X}_2^0\) is defined similarly.
Example 2.1. Let \(X_1\) and \(X_2\) be random variables with a joint pdf \[f\big (X_1,X_2\big ) = \begin {cases} 1 + \alpha (2x_1-1)(2x_2-1), & 0<x_1,x_2<1\quad -1\leq \alpha \leq 1\\\\ 0, & \text {otherwise}\\ \end {cases} \]
- 1.
- Is \(f\big (X_1,X_2\big )\) a density function?
- 2.
- If yes, what is
- (a)
- \(f_1\big (X_1\big )\)?
- (b)
- \(f_2\big (X_2\big )\)?
Solution.
- 1.
- \begin {align*} \int ^1_0\int ^1_0f\big (X_1,X_2\big ) dX_1dX_2 & = \int ^1_0\int ^1_0 \big (1+\alpha (2x_1-1)(2x_2-1)\big )dx_1dx_2\\ & = \int ^{1,1}_{0,0} dx_1dx_2 + \alpha \int ^{1,1}_{0,0}(2x_1-1)(2x_2-1)dx_1dx_2\\ & = 1 + \alpha \int ^1_0 (2x_2-1)(x^2_1-x_1)\big |^1_0dx_2\\ & = 1 + \alpha \int ^1_0(2x_2-1)(0)dx_2\\ & = 1\\ \end {align*}
- 2.
-
- (a)
- \(\displaystyle {f_1\big (X_1\big ) = \int _{x_2}f(x_1,x_2)dx_2}= 1\quad 0<x_1<1\)
- (b)
- \(\displaystyle {f_2\big (X_2\big ) = \int _{x_1} f\big (x_1,x_2\big )dx_1= \begin {cases} 1, & 0<x_2 < 1\\\\ 0, & \text {Otherwise}\\ \end {cases} }\)
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