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January 24th, 2018, 12:00 AM  #1 
Newbie Joined: Jan 2018 From: ChengDu China Posts: 4 Thanks: 0  a strange joint probability problem
For example, F (x, y) is a twodimensional joint probability distribution function, where X is a normal normal distribution, Y is 01 binomial distribution with parameter q. How to find the expectation of F (x, y)? thanks for solving this problem Last edited by fun; January 24th, 2018 at 12:17 AM. Reason: make the question more clearly 
January 24th, 2018, 08:38 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,009 Thanks: 1042 
your question doesn't make sense as written. $F(x,y)$ is not a random variable and thus it doesn't have an expectation. you can find $E[X]$, or $E[Y]$, or $E[g(X,Y)]$ as $X$ and $Y$ are random variables So what exactly are you trying to find? 
January 24th, 2018, 04:54 PM  #3  
Newbie Joined: Jan 2018 From: ChengDu China Posts: 4 Thanks: 0  Quote:
that's my fault, I will make the question clearer！ suppose X,Y are both random variables F(X,Y) is a function on X Y X obeys the normal distribution with mean m, standard deviation n Y obeys poisson distribution with parameter p so how to caculate E[F(X,Y)]  
January 24th, 2018, 07:12 PM  #4  
Senior Member Joined: Sep 2015 From: USA Posts: 2,009 Thanks: 1042  Quote:
let $Z=F(X,Y)$ $\large E[Z] = \displaystyle \sum \limits_{y=0}^\infty~\int_{\infty}^{\infty}~F(x,y) \dfrac{1}{\sqrt{2\pi}n}e^{\frac{(xm)^2}{2n^2}}\cdot \dfrac{p^y e^{p}}{y!}~dx $ This might simplify a little bit without specifying $F(X,Y)$ but not much.  
January 24th, 2018, 11:13 PM  #5  
Newbie Joined: Jan 2018 From: ChengDu China Posts: 4 Thanks: 0  Quote:
you mean that to solve this kind of question we don't need use Lebesgue integral，Riemann integral is enough?  
January 25th, 2018, 07:32 AM  #6  
Senior Member Joined: Sep 2015 From: USA Posts: 2,009 Thanks: 1042  Quote:
I suppose you could come up with one that would require a Lebesgue integral but in general you won't need one. Each value of $Y$ provides, depending on $F(x,y)$ an Riemann integrable function in $x$  
January 25th, 2018, 04:54 PM  #7  
Newbie Joined: Jan 2018 From: ChengDu China Posts: 4 Thanks: 0  Quote:
can you give me an example that we need to use Lebsgue intergral? Let$F[X,Y]=tan(X+Y)$ can we use Riemann intergral?  

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calculate, distribution, expectation, joint, probability 
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