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2. Let Z1 and Zo be independent standard normal random variables. Let! X= 221 +372 +12...
| Assume that Z1 and Z2 are two independent random variables that follow the standard normal dist ribution N(0,1), so that each of them has the density 1 (z) ooz< oo. e '2т X2 X2+Y2 Let X 212,Y 2Z1 2Z2, S X2Y2, and R (a) Please find the joint density of (Z1, Z2). (b) From (a), please find the joint density of (X,Y) (c) From (b), please find the marginal densit ies of X and Y. (d) From (b) and...
Let X1 and X2 be two independent standard normal random variables. Define two new random variables as follows: Y-Xi X2 and Y2- XiBX2. You are not given the constant B but it is known that Cov(Yi, Y2)-0. Find (a) the density of Y (b) Cov(X2, Y2)
Let (X1, Y1) and (X2, Y2) be independent and identically distributed continuous bivariate random variables with joint probability density function: fX,Y (x,y) = e-y, 0 <x<y< ; =0 , elsewhere. Evaluate P( X2>X1, Y2>Y1) + P (X2 <X1, Y2<Y1) .
Let X1, X2, X3 be independent Binomial(3,p) random variables. Define Y1 = X1 + X3 and Y2 = X2 + X3. Define Z1 = 1 if Y1 = 0; and 0 otherwise. Define Z2 = 1 if Y2 = 0; and 0 otherwise. As Z1 and Z3 both contain X3, are Z1 and Z3 independent? What is the marginal PMF of Z1 and Z2 and joint PMF of (Z1, Z2) and what is the correlation coefficient between Z1 and Z2?
5. Let X1 and X2 be two independent standard normal random variables. Define two new random variables as follows: Yı = X1 + X2 and ½ = X1 + ßX2. You are not given the constant β but it is known that Cov(Yi,Y) = 0. Find (a) the density of Y2 (b) Cov(Xy½),
Assume that Z1 and 22 are two independent random variables that follow the standard normal distribution N(0,1), so that each of them has the density 0(3) = , Let X = {{z + 12 Zz, Y = 122- x2z2, S = x2 + y2, and R= * Answers, a,b,c,d,e are provided below need help with g, hi (g) From (e), please find the density of (X,Y) (note that X2 and Y2 are independent from (a)). (h) From (g), please find...
2. Let the random variables Y1 and Y, have joint density Ayſy22 - y2) 0<yi <1, 0 < y2 < 2 f(y1, y2) = { otherwise Stom.vn) = { isiml2 –») 05451,05 ms one a independent, amits your respon a) Are Y1 and Y2 independent? Justify your response. b) Find P(Y1Y2 < 0.5). on the
PROB 4 Let Xi and X2 be independent exponential random variables each having parameter 1 i.e. fx(x) = le-21, x > 0, (i = 1,2). Let Y1 = X1 + X2 and Y2 = ex. Find the joint p.d.f of Yi and Y2.
Unif (0, 1) 5. Suppose U1 and U2 i= 1,2. Let X; = - log(1 - U;), i = 1,2. [0, 1], U are independent uniform random variables on (a) Show that X1 and X2 are independent exponential random variables with mean 1, X; ~ Еxp(1), і — 1,2. (b) Find the joint density function of Y1 = X1 + X2 and Y2 = X1/X2 and show that Y1 and Y2 are independent. Unif (0, 1) 5. Suppose U1 and...
please help me! 4. Let Xi and X2 be two independent standard normal random variables Define two new random variables as follows: Yǐ = X1 +X2 and Y2 = X1 +ßX2. Y t ß but it is known that Cor(Y,Y)-0. Find ou are not given the const an (i) The density of Y2 . (ii) Cov(X2, Y2). (your answers shouldn't involve β)