Double Integral Limits

[ku]

How do you know what limits to put when double integrating.

 

For example for the joint probability distribution function

 

[math]f(x,y)= \begin{cases}

2 & \mbox{if } 0 \leq y \leq x \leq 1\\

0 & \mbox{elsewhere}

\end{cases}

[/math]

 

we want to calculate the marginal probability density function of X which is

 

[math]f(x)=\int_{-\infty}^{\infty}f(x,y)dy = \int_{0}^{x}2 dy = 2x[/math].

 

How come 0 and x are used as limits? Why couldn't we have picked, say, 0 and 1?

[swansont]

How do you know what limits to put when double integrating.

 

For example for the joint probability distribution function

 

[math]f(x' date='y)= \begin{cases}

2 & \mbox{if } 0 \leq y \leq x \leq 1\\

0 & \mbox{elsewhere}

\end{cases}

[/math']

 

we want to calculate the marginal probability density function of X which is

 

[math]f(x)=\int_{-\infty}^{\infty}f(x,y)dy = \int_{0}^{x}2 dy = 2x[/math].

 

How come 0 and x are used as limits? Why couldn't we have picked, say, 0 and 1?

 

Because the value of the function depends on whether y>x or not

 

If the functional value had depended on a hard value for y, you would integrate to that limit.