Homework Answers
Note that E(aX+bY+cZ)= aE(X)+bE(Y)+cE(Z)
and V(aX+bY+cZ)= a2V(X)+b2V(Y)+c2V(Z)+2ab*cov(X,Y)+2bc*cov(Y,Z)+2ac*cov(X,Z).
Using these formulaes we get,
If
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purcased, 1, äre +o an+esectivery Ie almount of lype II chemical purchased, Y2, has E(Y) =...
3. (30pt) Suppose that E(Y) = 1, E(Y2) = 2, E(Y3) = 3, V(Y1) = 6,...
3. (30pt) Suppose that E(Y) = 1, E(Y2) = 2, E(Y3) = 3, V(Y1) = 6, V(Y2) = 7,V (Y3) = 8, Cov(Yı, Y2) = 0, Cov(Yı, Y3) = -4 and 10 1 2 3 Cov(Y2, Y3) = 5. Also define a = 20 and A = 4 5 6 30/ ( 7 8 9 (a) (10pt) Find the expected value and variance covariance matrix of Y, where Y = Y2 (b) (10pt) Compute Eſa'Y) and E(AY). (c) (10pt) Compute...
4. A random point (X, Y ) is chosen uniformly from within the unit disk in R2, {(x, y)|x2+y2< 1} (a) Let (R, O) deno...
4. A random point (X, Y ) is chosen uniformly from within the unit disk in R2, {(x, y)|x2+y2< 1} (a) Let (R, O) denote the polar coordinates of the point (X,Y). Find the joint p.d.f. of R and . Compute the covariance between R and 0. Are R and e are independent? (b) Find E(XI{Y > 0}) and E(Y|{Y > 0}) (c) Compute the covariance between X and Y, Cov(X,Y). Are X and Y are independent?
4. A random...
3. F =< y2 + x-3, 2xy + e? -y + 2, ye? +2z -4> (1)...
3. F =< y2 + x-3, 2xy + e? -y + 2, ye? +2z -4> (1) Prove or disprove that F is conservative. (ii) If F is conservative find the potential function f.
(1) A supplier of kerosene has a weekly demand Y possessing a probability density function given ...
having troubles with a (ii) and (c). thanks!
(1) A supplier of kerosene has a weekly demand Y possessing a probability density function given by 0, elsewhere with measurements in hundreds of gallons. The suppliers profit is given by U-10Y-4. (The c.d.f. was calculated in Tutorial Question 4 of week 3) (a) Find the p.d.f. for U i) using the distribution method and) the trans- (b) Use the answer to part (a), to find E(U) (using the p.d.f of U)...
Given the potential V=x^yz2 [V] find the electric field E at (x 1,y=2,z=1 (i) (ii) calculate...
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Question 1 Consider the following Multiple Regression Model yı BoB1B2 + El, y2 BIB2E2 y3 B2Es,...
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Problems binomial random variable has the moment generating function ψ(t)-E( ur,+1-P)". Show, that EIX) np and...
Problems binomial random variable has the moment generating function ψ(t)-E( ur,+1-P)". Show, that EIX) np and Var(X)-np(1-P) using that EXI-v(0) and Elr_ 2. Lex X be uniformly distributed over (a b). Show that EX]- and Varm-ftT using the first and second moments of this random variable where the pdf of X is () Note that the nth i of a continuous random variable is defined as E (X%二z"f(z)dz. (z-p?expl- ]dr. ơ, Hint./ udv-w-frdu and r.e-//agu-VE. 3. Show that 4 The...
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P...
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P(X>x) P(x (3 = Similarly, for all y [0,1,2,...^, show that Show your working only for one of the two identities that are pre- sented above. Hint: You may use the following identity without proving it. For any non-negative integer (, we have:...
Consider 1-2 Vr? + y + 3 LLL da dydar. V1-38-98 V +y + y2 +22...
Consider 1-2 Vr? + y + 3 LLL da dydar. V1-38-98 V +y + y2 +22 +y +22-2 the origin to the point (2, y, ) makes with the z-axis is a new angle which we will label o, and we label the length of the line segment p. We can now determine the remaining side-lengths of our new triangle. Let us try to label our point (2, y, z) in only p and 6. Our labeled triangle gives us...
3. (30pt) Suppose that E(Y) = 1, E(Y2) = 2, E(Y3) = 3, V(Y1) = 6, V(Y2) = 7,V (Y3) = 8, Cov(Yı, Y2) = 0, Cov(Yı, Y3) = -4 and 10 1 2 3 Cov(Y2, Y3) = 5. Also define a = 20 and A = 4 5 6 30/ ( 7 8 9 (a) (10pt) Find the expected value and variance covariance matrix of Y, where Y = Y2 (b) (10pt) Compute Eſa'Y) and E(AY). (c) (10pt) Compute...
4. A random point (X, Y ) is chosen uniformly from within the unit disk in R2, {(x, y)|x2+y2< 1} (a) Let (R, O) denote the polar coordinates of the point (X,Y). Find the joint p.d.f. of R and . Compute the covariance between R and 0. Are R and e are independent? (b) Find E(XI{Y > 0}) and E(Y|{Y > 0}) (c) Compute the covariance between X and Y, Cov(X,Y). Are X and Y are independent?
4. A random...
3. F =< y2 + x-3, 2xy + e? -y + 2, ye? +2z -4> (1) Prove or disprove that F is conservative. (ii) If F is conservative find the potential function f.
having troubles with a (ii) and (c). thanks!
(1) A supplier of kerosene has a weekly demand Y possessing a probability density function given by 0, elsewhere with measurements in hundreds of gallons. The suppliers profit is given by U-10Y-4. (The c.d.f. was calculated in Tutorial Question 4 of week 3) (a) Find the p.d.f. for U i) using the distribution method and) the trans- (b) Use the answer to part (a), to find E(U) (using the p.d.f of U)...
Given the potential V=x^yz2 [V] find the electric field E at (x 1,y=2,z=1 (i) (ii) calculate the work done in moving a 2 uC charge from A-(1,1,1) to B-(4,-1,1)
Question 1 Consider the following Multiple Regression Model yı BoB1B2 + El, y2 BIB2E2 y3 B2Es, and y4 Bo+BI4 Suppose that & 's are independent and identically distributed N(0, o2 ) a) Write down the model in the matrix form b) Show that 2 2 1 X'X2 3 2 1.67 -1.33 0.33 (X'X) 1.67 Note that -1.33 -0.67 1 2 3 0.33 -0.67 0.67 c) Find unbiased estimators for Bo, Bi, and B2 given that y 3, y2 1, y3-...
problems binomial random, veriable has the moment generating function, y(t)=E eux 1. A nd+ 1-p)n. Show that EIX|-np and Var(X) np(1-p) using that EIX)-v(0) nd E.X2 =ψ (0). 2. Lex X be uniformly distributed over (a b). Show that ElXI 쌓 and Var(X) = (b and second moments of this random variable where the pdf of X is (x)N of a continuous randonn variable is defined as E[X"-广.nf(z)dz. )a using the first Note that the nth moment 3. Show that...
Problems binomial random variable has the moment generating function ψ(t)-E( ur,+1-P)". Show, that EIX) np and Var(X)-np(1-P) using that EXI-v(0) and Elr_ 2. Lex X be uniformly distributed over (a b). Show that EX]- and Varm-ftT using the first and second moments of this random variable where the pdf of X is () Note that the nth i of a continuous random variable is defined as E (X%二z"f(z)dz. (z-p?expl- ]dr. ơ, Hint./ udv-w-frdu and r.e-//agu-VE. 3. Show that 4 The...
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P(X>x) P(x (3 = Similarly, for all y [0,1,2,...^, show that Show your working only for one of the two identities that are pre- sented above. Hint: You may use the following identity without proving it. For any non-negative integer (, we have:...
Consider 1-2 Vr? + y + 3 LLL da dydar. V1-38-98 V +y + y2 +22 +y +22-2 the origin to the point (2, y, ) makes with the z-axis is a new angle which we will label o, and we label the length of the line segment p. We can now determine the remaining side-lengths of our new triangle. Let us try to label our point (2, y, z) in only p and 6. Our labeled triangle gives us...
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