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EE 503: Problem Set #10 Due Monday Apr. 12, 2021 (9pm)

a）找到CDF FY（y）和PDF fY（y）。
b）找到条件CDF FY | X = 1（y）和条件PDF fY | X = 1（y）。 c）查找P [X = 1 | Y≥1]
d）求出P􏰂（X + Y）2≤1 /2􏰃。
e）找出E [Y]和Var（Y）。

b）对于y> 0且x̸= 0，通过使用（X，R）→（X，Y）的PDF转换找到fX | Y（x | y）以找到fX，Y（x，y）。

a）通过使用FY | X = x（y）= P [Y≤y| X = x]查找fX | Y（x | y）。请记住分别处理x> 0和x <0的情况。 b）通过使用（X，R）→（X，Y）的PDF转换来找到fX | Y（x | y）以找到fX，Y（x，y）。

IV。高斯的线性组合

fX（x）= 1，-∞<x <∞π（1 + x2）

X = -1的p值更有可能。

I. INDEPENDENT ADDITIVE GAUSSIAN NOISE

Let X and N be independent random variables. Suppose X is uniform over {-1,0,1} (equally likely over all 3 options). SupposeN isGaussianN􏰀02􏰁.DefineY =X+N.

a) Find the CDF FY (y) and PDF fY (y).
b) Find the conditional CDF
FY |X=1(y) and the conditional PDF fY |X=1(y). c) Find P[X =1|Y 1]
d) Find P 􏰂(X +Y)2 1/2􏰃.
e) Find
E[Y ] and Var(Y ).

II. COMPUTING fX|Y (x|y)

Let Y = X2R where X,R are independent with X Gaussian N(0,1) and R exponential with parameter λ > 0 a)Fory>0andx̸=0,findfX|Y(x|y)byusingFY|X=x(y)=P[Y y|X=x].
b) For
y > 0 and x ̸= 0, find fX|Y (x | y) by using the PDF transformation for (X, R) (X, Y ) to find fX,Y (x, y).

III. ANOTHER COMPUTATION OF fX|Y (x|y)

Let Y = RX where R, X are i.i.d Gaussian N (0, 1).
a)Find
fX|Y(x|y)byusingFY|X=x(y)=P[Y y|X=x].Remembertotreatcasesx>0andx<0separately. b) Find fX|Y (x | y) by using the PDF transformation for (X, R) (X, Y ) to find fX,Y (x, y).

IV. LINEAR COMBINATION OF GAUSSIAN

Let X, Y, Z be mutually independent Gaussian random variables. Assume X has mean 1 and variance 2, Y has mean 0 and variance 1,Z has mean 8 and variance 1/2. Let W = Z 2X +Y/4. Compute the PDF fW(w) for all w R. Use the Q() function to find P [W > 9].

Suppose that X as the following PDF:

V. PDF TRANSFORMATION METHOD

fX (x) = 1 , −∞ < x < π(1+x2)

Let Y = a . Find the PDF of Y using PDF transformation. 1+X2

VI. ANOTHER PDF TRANSFORMATION PROBLEM Suppose that X follows a U [2, 1] distribution. Let Y = 2X 2 1. Find the PDF of Y .

VII. BOOK PROBLEM 5.35
For part (c), your answer should depend on the value of
p. Find out the values of p for which X = 1 is more likely and

the values of p for which X = 1 is more likely.

VIII. BOOK PROBLEM 5.101
IX. B
OOK PROBLEM 4.92 (A) AND (B)

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