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January 24th, 2018, 01:00 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 two-dimensional joint probability distribution function, where X is a normal normal distribution, Y is 0-1 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 01:17 AM. Reason: make the question more clearly |
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January 24th, 2018, 09:38 AM | #2 |
Senior Member Joined: Sep 2015 From: USA Posts: 2,299 Thanks: 1220 |
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? |
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January 24th, 2018, 05:54 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)] | |
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January 24th, 2018, 08:12 PM | #4 | |
Senior Member Joined: Sep 2015 From: USA Posts: 2,299 Thanks: 1220 | 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{(x-m)^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. | |
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January 25th, 2018, 12:13 AM | #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? | |
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January 25th, 2018, 08:32 AM | #6 | |
Senior Member Joined: Sep 2015 From: USA Posts: 2,299 Thanks: 1220 | 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 $x$ | |
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January 25th, 2018, 05:54 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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