Math  /  Data & Statistics

Question2. Continue to love Gaussian: XX and YY are independent identically distributed (i.i.d.) Gaussian, i.e., XN(μ,σ2)X \sim N\left(\mu, \sigma^{2}\right), YN(μ,σ2)Y \sim N\left(\mu, \sigma^{2}\right). (a) ( 5pts\mathbf{5} \mathbf{p t s} ) Define Z=2X+3YZ=\sqrt{2} X+\sqrt{3} Y, find fZ(z)f_{Z}(z). (b) ( 10 pts)\mathbf{1 0} \mathbf{~ p t s )} Set μ=0,σ2=1\mu=0, \sigma^{2}=1. Find P(5Z5)P(\sqrt{5} \leq Z \leq 5). Leave your answer in terms of Q()Q(\cdot) function.
Recall: Q(x)=x12πet22dtQ(x)=\int_{x}^{\infty} \frac{1}{\sqrt{2 \pi}} e^{-\frac{t^{2}}{2}} d t. (c) ( 10pts)\mathbf{1 0} \mathbf{p t s )} Suppose now we have XX and YY with the same Gaussian marginals fX(x)f_{X}(x) and fY(y)f_{Y}(y), and μ=0,σ2=1\mu=0, \sigma^{2}=1. Further, we are told that XX and YY are jointly Gaussian and CXY=12C_{X Y}=\frac{1}{2}. Find fXY(xy)f_{X \mid Y}(x \mid y). How does your answer compare with the i.i.d. case stated in the original question above with μ=0,σ2=1\mu=0, \sigma^{2}=1 ?

Studdy Solution
Given X X and Y Y are jointly Gaussian with correlation CXY=12 C_{XY} = \frac{1}{2} .
The conditional distribution of XY=y X \mid Y = y is Gaussian:
XY=yN(μX+σXYσY2(yμY),σX2(1ρ2)) X \mid Y = y \sim N\left(\mu_X + \frac{\sigma_{XY}}{\sigma_Y^2}(y - \mu_Y), \sigma_X^2(1 - \rho^2)\right)
With μ=0\mu = 0, σ2=1\sigma^2 = 1, and ρ=12\rho = \frac{1}{2}:
XY=yN(12y,34) X \mid Y = y \sim N\left(\frac{1}{2}y, \frac{3}{4}\right)
In the i.i.d. case, X X and Y Y are independent, so:
XY=yN(0,1) X \mid Y = y \sim N(0, 1)
Comparison: In the i.i.d. case, the conditional mean is 0 and the variance is 1, whereas in the correlated case, the mean is 12y\frac{1}{2}y and the variance is 34\frac{3}{4}.
The solution to each part is as follows: (a) ZN((2+3)μ,5σ2) Z \sim N((\sqrt{2} + \sqrt{3})\mu, 5\sigma^2) (b) P(5Z5)=Q(1)Q(5) P(\sqrt{5} \leq Z \leq 5) = Q(1) - Q(\sqrt{5}) (c) XY=yN(12y,34) X \mid Y = y \sim N\left(\frac{1}{2}y, \frac{3}{4}\right)

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