O. Let X1 and X2 be two random variables, and let Y = (X1 + X2)2. Suppose that E[Y ] = 25 and that the variance of X1 and X2 are 9 and 16, respectively.
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TOPIC:Covariance and Correlation between random variables.
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O. Let X1 and X2 be two random variables, and let Y = (X1 + X2)2....
15. Let the random variables X1 and X2 be the payoffs of two different Suppose E(X;)...
15. Let the random variables X1 and X2 be the payoffs of two different Suppose E(X;) = E(X2) = 100, and V( X) = V(X;) = investments 10. Suppose an investor owns 50% of each investment so the total payoff is: (X1+ X2 ) /2. There is a fixed fee (brokerage fee, for example) of 15 to acquire the two investments. So the investor's net payoff is: (Xi+X2)/2 - 15 a) Is the investor's net payoff a random variable? If...
Let X1 and X2 be independent random variables with means μ1 and μ2, and variances σ21...
Let X1 and X2 be independent random variables with means μ1 and μ2, and variances σ21 and σ22, respectively. Find the correlation of X1 and X1 + X2. Note that: The covariance of random variables X; Y is dened by Cov(X; Y ) = E[(X - E(X))(Y - E(Y ))]. The correlation of X; Y is dened by Corr(X; Y ) =Cov(X; Y ) / √ Var(X)Var(Y )
Given random variables X1, X2, Y with E[Y | X1, X2] = 5X1 + X1X2 and...
Given random variables X1, X2, Y with E[Y | X1, X2] =
5X1 + X1X2 and E[Y
2
| X1, X2] =
25X2
1X2
2 + 15, find
E[(X1Y + X2)
2
| X1, X2].
ㄨ竺Bin(2.1/4). Suppose X and Y are independent random variables. Find the expected value of YX. Hnt: Consider conditioning on the events (X-j)oj0,1,2. 9. Given random variables XI,X2, Y with E'Y | XiN;|-5X1 + X1X2 and Ep2 1 X1,X2] 25XX15, find 10. Let X and Y...
Let X1 and X2 be two independent continuous random variables. Define and S-Ixpo+2xso) where Ry and...
Let X1 and X2 be two independent continuous random variables. Define and S-Ixpo+2xso) where Ry and R2 are the Wilcoxon signed ranks of X, and X2, respectively. (a) Assume that X, and X2 have symmetric distributions about 0. Show that Pr(T ) Pr(S-t) for 0,1,2,3 using the properties of symmetry:-Xi ~ x, and Pr(X, > 0)-Pr(X, <0) = 0.5 (b) Suppose that X1 and X2 are identically distributed with common density -05%:- 10.5sx <0 0.5 0sxs1 show that Pr(T+-): Pr(S...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y)...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y;...
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y; be the partial sums, i.e., Y1 = X1, Y2 = X1 + X2, Y3 = X1 + X2 + X3. 1. What is the distubution for each Yį, i = 1, 2, 3? 2. What is the expected value for Y1 + Y2 +Yz? 3. Are Yį and Y2 independent? Explain it by computing their joint P.M.F. 4. What is the variance of Y1...
Let Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter p. Suppose that Y, X1 and X2 are independent. Proof u...
Let Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter p. Suppose that Y, X1 and X2 are independent. Proof using the de finition of distribution function that the the distribution function of Z =Y Xit(1-Y)X2 is F = pF14(1-p)F2 Don't use generatinq moment functions, characteristic functions) Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter...
. Let X1, X2, X3 be random variables each with distributions given by the pdf там,...
. Let X1, X2, X3 be random variables each with distributions given by the pdf там, 53-1 Also, the correlation coefficients of X1, X2, x3 are ρί/:-03,P13-0.6, p23-0.5. Find the mean and variance of
(2) Given two independent variables X1 and X2 having Bernoulli distribution with parameter p=1/3, let Y1...
(2) Given two independent variables X1 and X2 having Bernoulli distribution with parameter p=1/3, let Y1 = 2X1 and Y2 = 2X2. Then A E[Y1 · Y2] = 2/9 BE[Y1 · Y2] = 4/9 C P[Y1 · Y2 = 0) = 1/9 D P[Y1 · Y2 = 0) = 2/9 (3) Let X and Y be two independent random variables having gaussian (normal) distribution with mean 0 and variance equal 2. Then: A P[X +Y > 2] > 0.5 B...
Let X1, X2, and X3 be uncorrelated random variables, each with 4. (10 points) Let Xi,...
Let X1, X2, and X3 be uncorrelated random variables, each
with
4. (10 points) Let Xi, X2, and X3 be uncorrelated random variables, each with mean u and variance o2. Find, in terms of u and o2 a) Cov(X+ 2X2, X7t 3X;) b) Cov(Xrt X2, Xi- X2)
15. Let the random variables X1 and X2 be the payoffs of two different Suppose E(X;) = E(X2) = 100, and V( X) = V(X;) = investments 10. Suppose an investor owns 50% of each investment so the total payoff is: (X1+ X2 ) /2. There is a fixed fee (brokerage fee, for example) of 15 to acquire the two investments. So the investor's net payoff is: (Xi+X2)/2 - 15 a) Is the investor's net payoff a random variable? If...
Given random variables X1, X2, Y with E[Y | X1, X2] =
5X1 + X1X2 and E[Y
2
| X1, X2] =
25X2
1X2
2 + 15, find
E[(X1Y + X2)
2
| X1, X2].
ㄨ竺Bin(2.1/4). Suppose X and Y are independent random variables. Find the expected value of YX. Hnt: Consider conditioning on the events (X-j)oj0,1,2. 9. Given random variables XI,X2, Y with E'Y | XiN;|-5X1 + X1X2 and Ep2 1 X1,X2] 25XX15, find 10. Let X and Y...
Let X1 and X2 be two independent continuous random variables. Define and S-Ixpo+2xso) where Ry and R2 are the Wilcoxon signed ranks of X, and X2, respectively. (a) Assume that X, and X2 have symmetric distributions about 0. Show that Pr(T ) Pr(S-t) for 0,1,2,3 using the properties of symmetry:-Xi ~ x, and Pr(X, > 0)-Pr(X, <0) = 0.5 (b) Suppose that X1 and X2 are identically distributed with common density -05%:- 10.5sx <0 0.5 0sxs1 show that Pr(T+-): Pr(S...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y; be the partial sums, i.e., Y1 = X1, Y2 = X1 + X2, Y3 = X1 + X2 + X3. 1. What is the distubution for each Yį, i = 1, 2, 3? 2. What is the expected value for Y1 + Y2 +Yz? 3. Are Yį and Y2 independent? Explain it by computing their joint P.M.F. 4. What is the variance of Y1...
. Let X1, X2, X3 be random variables each with distributions given by the pdf там, 53-1 Also, the correlation coefficients of X1, X2, x3 are ρί/:-03,P13-0.6, p23-0.5. Find the mean and variance of
(2) Given two independent variables X1 and X2 having Bernoulli distribution with parameter p=1/3, let Y1 = 2X1 and Y2 = 2X2. Then A E[Y1 · Y2] = 2/9 BE[Y1 · Y2] = 4/9 C P[Y1 · Y2 = 0) = 1/9 D P[Y1 · Y2 = 0) = 2/9 (3) Let X and Y be two independent random variables having gaussian (normal) distribution with mean 0 and variance equal 2. Then: A P[X +Y > 2] > 0.5 B...
Let X1, X2, and X3 be uncorrelated random variables, each
with
4. (10 points) Let Xi, X2, and X3 be uncorrelated random variables, each with mean u and variance o2. Find, in terms of u and o2 a) Cov(X+ 2X2, X7t 3X;) b) Cov(Xrt X2, Xi- X2)
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