Calculus Calculus Math Forum

 August 1st, 2016, 07:35 AM #1 Newbie   Joined: Aug 2016 From: UK Posts: 2 Thanks: 0 Sum over random vector values I have the following expression: $\sum_{\mathbf{x}}e^{\mathbf{x}}$, where $\mathbf{x}$ is a random vector (i.e., $\mathbf{x} \in \mathbb{R}^D$ ), and $\sum_{\mathbf{x}}$ represents a sum over all possible values that $\mathbf{x}$ can assume. Now, I'm trying to rewrite this expression more explicitly, but I'm a bit stuck and I'm not sure if I'm doing it correctly, here are two of my attempts: $\sum_{\mathbf{x}}e^{\mathbf{x}} = \sum_{i}^{D} \int e^{x_i} \mathop{dx_i}$ or $\sum_{\mathbf{x}}e^{\mathbf{x}} = \int e^{\sum_{i}^{D} x_i} \mathop{dx_i} = \prod_{i}^{D}\int e^{x_i} \mathop{dx_i}$ I'm more biased toward the second, but again, I'm not really sure that it is correct, and I'm even doubtful about the correctness of the notation that I'm using (because in the integral $x_i$ depends on $i$ which is the index of the summation, and this doesn't seem entirely right to me). August 1st, 2016, 03:02 PM #2 Global Moderator   Joined: May 2007 Posts: 6,770 Thanks: 700 Your original expression doesn't make sense. The vector has a continuum of values and you can't sum. You might consider looking at it as some sort of integral. August 2nd, 2016, 01:19 AM #3 Newbie   Joined: Aug 2016 From: UK Posts: 2 Thanks: 0 Thanks mathman, I think you're right, the notation of my original expression is wrong in the first place. However, considering that $\mathbf{x} \in \mathbb{R}^D$, if I had my original expression written directly like $\prod_{i}^{D}\int e^{x_i} \mathop{dx_i}$, i.e., with an explicit integral for each random variable and a product from $1$ to $D$, I think it should be correct, right? To add more details, my intention was to use a more compact notation for the following expression: $\int e^{x_1} \mathop{dx_1}\times\int e^{x_2} \mathop{dx_2}\times\dots\times\int e^{x_D} \mathop{dx_D}$ The only way I can think of is: $\prod_{i}^{D}\int e^{x_i} \mathop{dx_i}$ Is there a better way to write the same? August 2nd, 2016, 03:01 PM #4 Global Moderator   Joined: May 2007 Posts: 6,770 Thanks: 700 Not really. Just be careful for the product notation, i ranges from 1 to D. Tags notation, random, random variable, sum, summation, values, vector Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post messaoud2010 Probability and Statistics 1 July 18th, 2014 11:26 AM Santik Advanced Statistics 2 July 5th, 2013 12:52 PM ranjan Linear Algebra 1 January 19th, 2012 01:11 PM rmas Advanced Statistics 1 January 1st, 2010 12:57 PM begyu85 Algebra 3 February 19th, 2008 01:34 PM

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