[AI/ML Examples] Factorization criterion in action in the special case of the bivariate normal pdf

Factorization criterion in action in the special case of the bivariate normal pdf

포스트 난이도: HOO_Senior

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# Example 1

 

Find the marginals (i.e., the marginal pdfs of Xand Y from the joint pdf). If you are unable to do this analytically (which is fine, nopenalties), assume μX=μY= 0,σX=σY= 1, and ρ= 0.5; specifically, use numerical integration to find the values of the marginal pdfs on a fine grid from on the interval [-3,3], plot those and compare withNormal(0,1) pdf

f _y(y)=\int_{-\infty }^{\infty }f_x_y(x,y)dx = \frac{1}{2\pi \sigma _x\sigma _y\sqrt{1-p^{2}}}\int_{-\infty }^{\infty }e^{-\frac{1}{2}Q(x,y)}dy

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Q(x,y)=\frac{(\frac{x-\mu _x}{\sigma_x})^{2}-2p(\frac{x-\mu _x}{\sigma _x})(\frac{y-\mu _y}{\sigma _y})+(\frac{y-\mu _y}{\sigma _y})^{2}}{1-p^{2}}

=(\frac{x-a}{b})^{2}+c

a=\mu _x+p\frac{\sigma _x}{\sigma _y}(y-\mu )

b= \sigma _x\sqrt{1-p^{2}}

c=(\frac{y-\mu _y}{\sigma _y})^{2}

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f_y(y)=\frac{1}{2\pi \sigma _x\sigma _y\sqrt{1-p^{2}}}\int_{-\infty }^{\infty }e^{-\frac{1}{2}((\frac{x-a}{b})^{2}+c)^{2}}dx

f_y(y)=\frac{e^{-\frac{c}{2}}}{2\pi \sigma _x\sigma _y\sqrt{1-p^{2}}}\int_{-\infty }^{\infty }e^{-\frac{1}{2}((\frac{x-a}{b})^{2})}dx

f_Y(y)=\frac{e^{-\frac{c}{2}}}{\sqrt{2\pi} \sigma _x\sigma _y\sqrt{1-p^{2}}}\sigma _x\sqrt{1-p^{2}}

f_y(y)=\frac{1}{\sqrt{2\pi}\sigma _y}e^{-\frac{1}{2}(\frac{y-\mu _y}{\sigma _y})^{2}}

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# Example 2

 

Show that ifρ= 0then the rvs are independent. Marginally,X is normal with mean μX and variance σ2X(similarly, for Y).

 

If p=0,

f_x_y(x,y)=\frac{1}{2\pi \sigma _x\sigma _y}e^{-\frac{1}{2}((\frac{x-\mu _x}{\sigma _x})^{2}+(\frac{y-\mu _y}{\sigma _y})^{2})}

f_x(x)f_y(y)=\frac{1}{2\pi \sigma _x\sigma _y}e^{-\frac{1}{2}((\frac{x-\mu _x}{\sigma _x})^{2}+(\frac{y-\mu _y}{\sigma _y})^{2})}

f_x_y(x,y)=f_x(x)f_y(y)

Thus, X and Y are independent.

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# Example 3

 

Assume μX=μY= 0 and σX=σY= 1; let ρ be general (strictly between -1 and 1). For values of ρ on the grid from -0.75 to 0.75 with step size 0.25, p lot the conditional pdf of X given that Y= 1. If Y= 1 is the observed value, does the correlation (positive or negative) help one predicting X, relative to the case of ρ= 0? Briefly discuss.

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