Thread: How to determine pdf region

1. How to determine pdf region

Hi,

I have difficulty to understand the regions of integration for pdf, so here is some examples, I would appreciate if you can help:

So lets say we have $\displaystyle f(y_1, y_2)$, and we want to show that it is a valid pfd so:

$\displaystyle \int\int f(y_1,y_2) ~dy_1 ~dy_2 = 1$ Now:

Example 1: $\displaystyle 0 <= y_1 <= 2, 0<= y_2 <= 1, 2y_2 < y_1$

$\displaystyle \int_{0}^{2}\int_{0}^{y_1/2} f(x) ~dy_2 ~dy_1$

Example 2:

$\displaystyle y_1 -1 <= y_2 <= 1-y_1 , 0 <= y_1 <= 1$

$\displaystyle \int_{0}^{1}\int_{y_1 - 1}^{1-y_1} f(x) ~dy_2 ~dy_1$

Example 3:

$\displaystyle 0 <= y_1 <= y_2, y_1 + y_2 <= 2$

I don't have any idea here

thanks for the help

2. Originally Posted by hitman
Hi,

I have difficulty to understand the regions of integration for pdf, so here is some examples, I would appreciate if you can help:

So lets say we have $\displaystyle f(y_1, y_2)$, and we want to show that it is a valid pfd so:

$\displaystyle \int\int f(y_1,y_2) ~dy_1 ~dy_2 = 1$ Now:

Example 1: $\displaystyle 0 <= y_1 <= 2, 0<= y_2 <= 1, 2y_2 < y_1$

$\displaystyle \int_{0}^{2}\int_{0}^{y_1/2} f(x) ~dy_2 ~dy_1$

Example 2:

$\displaystyle y_1 -1 <= y_2 <= 1-y_1 , 0 <= y_1 <= 1$

$\displaystyle \int_{0}^{1}\int_{y_1 - 1}^{1-y_1} f(x) ~dy_2 ~dy_1$

Example 3:

$\displaystyle 0 <= y_1 <= y_2, y_1 + y_2 <= 2$

I don't have any idea here

thanks for the help
It would help if you posted the complete and unexpurgated version of eg. 3 that is in the book that you got it from.