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Consider the problem of approximating a continuous analog signal X with a discrete digital signal Y. To do this, we may divide the range of X into discrete intervals, [a0,a1), [a1,a2), etc. Then, if X takes on a value in the interval [a0,a1), we assign the value y1 to the discrete signal (predictor) Y, and so on for the other intervals.

Suppose that X is distributed on [0,1] according to the PDF f(x)=1.8x+0.2. If we discretize by picking a0=0, a1=1/2 and a2=1, how should we pick y1 and y2 to generate the best predictor of X (that is, to minimize E[(X−Y)^2]?

what is y1 y2 and what is the value of E[(X−Y)^2]

I suppose p(y1)=\int(1.8x+0.2)(0<x<1/2)

MJD
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Yihang
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  • Can you write $E[(X-Y)^2]$ as a function of $y_1$ and $y_2$ and then use calculus to minimize $\frac{dE}{dt_1}$ and $\frac{dE}{dt_2}$ ? – MJD Feb 26 '22 at 06:45
  • @MJD Thank you for your advice. I am wondering what the P(y1) and P(y2) are. Are them the integral of f(x) from (0, 1/2) (1/2, 1) respectively? – Yihang Feb 27 '22 at 02:34

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