For two Gaussian-distributed variables, $ Pr(X=x) = \frac{1}{\sqrt{2\pi}\sigma_0}e^{-\frac{(x-x_0)^2}{2\sigma_0^2}}$ and $ Pr(Y=y) = \frac{1}{\sqrt{2\pi}\sigma_1}e^{-\frac{(x-x_1)^2}{2\sigma_1^2}}$. What is probability of the case X > Y?
The probability of one Gaussian larger than another.
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probability
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1What do you know about $X-Y$ ? – 2012-08-03
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0If $X > Y$ what can you say about $X-Y$? – 2012-08-03
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3Are $X$ and $Y$ independent? And I don't agree when you write $Pr(X=x)=\dots$: the probability that a Gaussian random variable take a particular value is $0$ (but we can write $P(X\in A)=\int_A$ of the function you wrote). – 2012-08-03
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0Yes, do we have a name for this? – 2012-08-03
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0A name for *what*? – 2012-08-03
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0@Strin: [Probability density function](http://en.wikipedia.org/wiki/Probability_density_function)? – 2012-08-03
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0@DavideGiraudo: You are quite right, but this is a common abuse of notation. – 2012-08-03
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0@Nate: A common abuse of notation, to write $P(X=x)=f(x)$ to mean that $f$ is the density of $P_X$ with respect to Lebesgue measure? If you wish to indicate that we should not pay attention, I disagree. But this a too common **error**, yes. – 2012-08-03
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0@did: Many otherwise reputable textbooks use this notation deliberately. I personally don't like it either since it is, as you say, also a common error. But I just wanted to point out that someone who writes $P(X=x)$ for a density is not *necessarily* confused. – 2012-08-03
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0@Nate: *Many otherwise reputable textbooks use this notation deliberately*... OK. (But not in the part of the world where I live.) Any examples? – 2012-08-03