Say, for example, we have a $Y \sim \mathrm{Uniform}(0, 0.25)$, then $P(Y = y) = 4$ for all $Y \in [0, 0.25]$. My question is: what is the statistical significance/meaning of the $4$?
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2Quick note: $P(Y=y)=0$ (not $4$) for each $y\in[0,0.25]$. The density $f_Y(y)=4$. – Zen Aug 25 '13 at 18:48
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1Of course, sorry. – Christiaan Kruger Aug 25 '13 at 18:52
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It's the probability density. In this case, as the density is constant, integration of the density over the range from 0 to 0.25 is exactly equivalent to calculating the area of a rectangle: height of rectangle 4 $\times$ base of rectangle 0.25 equals 1, which is no more than saying that the integral of the probability density over the entire support equals the total probability.
Note: being cavalier about whether the support is (0,0.25) or [0, 0.25] makes no practical difference here, but being more careful would appeal to some.
Nick Cox
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Thank you. Yes I understand. But I'm trying to get an intuitive feeling for this. In a discrete distribution it's easy as the value at each point is the probability of that happening. In a continuous density that's not the case, as we know, we need to integrate. But does the value at a specific point represent anything? – Christiaan Kruger Aug 25 '13 at 18:39
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It doesn't. In fact, if you redefine the density arbitrarily at denumerably many points, the distribution of the random variable is still the same. – Zen Aug 25 '13 at 18:45