Expected Value and Multiple Integrals

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MuthuVeerappanR
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Expected Value and Multiple Integrals

Post by MuthuVeerappanR » Fri Jun 09, 2017 4:14 am

G'Morning all, I have a couple of questions regarding geometric probability.

Consider a point selected uniformly at random from a right angle triangle bounded by the lines $x=0$, $x=a$, and $y=mx$. Say I want to calculate the expected value of some function $f(x,y)$. Is it right to do it the following way?

$\mathbb{E}(f(x, y))=\displaystyle\frac{2}{ab}\int\limits_{0}^a\int\limits_{0}^{m x}f(x,y)\,dy\,dx$

Had it been a rectangle I would have no doubts about the integral. But in a right-angled triangle, this seems like doing the following.

Pick a value $x$ uniformly between $0$ and $a$. Now pick $y$ from $0$ to $mx$.

The second statement is where I have my doubts. Is it right to take that $y$ so selected will be uniform?

In cases like this, how do we check whether that particular value is uniformly distributed?

Say, If have $\mathbb{E}(f(x, y))$. Then is there a easier way to find any of $\mathbb{E}(f(a - x, y))$, $\mathbb{E}(f(x, m a - y))$, or $\mathbb{E}(f(a - x, m a - y))$??

Thanks for the help.

I'm not asking to this to solve any PE question. Also, I haven't solved any PE questions using these. So I don't think answering them wont be a spoiler.
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jaap
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Re: Expected Value and Multiple Integrals

Post by jaap » Fri Jun 09, 2017 8:24 am

I'm no expert on this, but I think it is correct. Basically, you have a sample space consisting of the points in the 2-dimensional plane $\Bbb R^2$, and a uniform Probability density function which has the value $\frac{2}{ab}$ for every point inside the triangle $T$, and $0$ outside of it:

$$pdf(P) =
\begin{cases}
\frac{2}{ab}, & \text{if $P \in T$} \\
0, & \text{if $P \notin T$}
\end{cases}$$

It is a valid pdf because its integral equals $1$:

$$ \int\limits_{P \in \Bbb R^2} pdf(P) dP = \int\limits_{P \in T} \frac{2}{ab} dP = \frac{2}{ab} \int\limits_{P \in T} 1 dP = \frac{2}{ab} \frac{ab}{2} = 1$$

The expected value of some function $f(P)$ is then the integral of the pdf times the function:

$$ \int\limits_{P \in \Bbb R^2} pdf(P) f(P) dP = \int\limits_{P \in T} \frac{2}{ab}f(P) dP $$

It does not matter how you do the integration over the triangle. Each part of the triangle contributes proportional to its area.

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Re: Expected Value and Multiple Integrals

Post by MuthuVeerappanR » Fri Jun 09, 2017 8:48 am

Thanks for the explanation jaap. After thinking about it today, I think I get that the points will be uniformly distributed. So kind of got clarity on the first question.

Again, anyone.. Any thoughts on the Second part of the question?? I don't think there is one but it surprises me that the analogy in single integral case is so simple to answer.
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Re: Expected Value and Multiple Integrals

Post by jaap » Fri Jun 09, 2017 2:08 pm

MuthuVeerappanR wrote:
Fri Jun 09, 2017 4:14 am
Say, If have $\mathbb{E}(f(x, y))$. Then is there a easier way to find any of $\mathbb{E}(f(a - x, y))$, $\mathbb{E}(f(x, m a - y))$, or $\mathbb{E}(f(a - x, m a - y))$??
No.
$\mathbb{E}(f(x, y)|(x,y) \in T)$ is the expected value that $f$ has within the triangle $T$, which is bounded by the lines $y=0$, $x=a$, and $y=mx$.

$\mathbb{E}(f(a-x, y)|(x,y) \in T) = \mathbb{E}(f(x, y)|(x,y) \in T_2)$ is the expected value that $f$ has within a triangle $T_2$, which is bounded by the lines $y=0$, $x=0$, and $y=m(a-x)$.

$\mathbb{E}(f(x, ma - y)|(x,y) \in T) = \mathbb{E}(f(x, y)|(x,y) \in T_3)$ is the expected value that $f$ has within a triangle $T_3$, which is bounded by the lines $y=ma$, $x=0$, and $y=ma-mx$.

$\mathbb{E}(f(a-x, ma - y)|(x,y) \in T) = \mathbb{E}(f(x, y)|(x,y) \in T_4)$ is the expected value that $f$ has within a triangle $T_4$, which is bounded by the lines $y=ma$, $x=a$, and $y=mx$.

These are different regions of the plane, so unless your function $f$ has some properties that relate the values over these triangles, there is not a lot you can do. However, $T \cup T_4 = T2 \cup T_3$, so once you know three of these you can deduce the fourth.

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Re: Expected Value and Multiple Integrals

Post by MuthuVeerappanR » Fri Jun 09, 2017 4:53 pm

That's nice jaap... Thanks a lot..
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