Derive the propagated errors SG or SGul for the following formulas (x, y, z denote variables...
Exercise 1 (1). X, Y are random variables (r.v.) and a,b,c,d are values. Complete the formulas using the expectations E(X), E(Y), variances Var(X), Var(Y) and covariance Cov(X, Y) (a) E(aX c) (b) Var(aX + c (d) Var(aX bY c) (e) The covariance between aX +c and bY +d, that is, Cov(aX +c,bY +d) f) The correlation between X, Y that is, Corr(X,Y (g) The correlation between aX +c and bY +d, that is, Corr(aX + c, bY +d)
If a and b are constants, the solution of (aye « +b)dx +(2ye xy + axy e ay – 1)dy=0 is a. Select one: ..y?e@y+bx = y b.y?(exy - 1) = 0 o c.yle axy +bx+y=C O d. ye xy + bx-y=C oe.yle ax + bx-y=C of.yle xy +bx-y=C g. e ary+bx= c o h. ye xy + x-y=C
5. Random variables X U[0, 1 and Y ~Exp(1) are independent (a) Compute P(X Y > z) for the case 0 S1 and the case z >1. b) Compute and plot the pdf of XY. (c) Give the MGF of X Y. 5. Random variables X U[0, 1 and Y ~Exp(1) are independent (a) Compute P(X Y > z) for the case 0 S1 and the case z >1. b) Compute and plot the pdf of XY. (c) Give the...
Let X and Y be independent identically distributed random variables with means µx and µy respectively. Prove the following. a. E [aX + bY] = aµx + bµy for any constants a and b. b. Var[X2] = E[X2] − E[X]2 c. Var [aX] = a2Var [X] for any constant a. d. Assume for this part only that X and Y are not independent. Then Var [X + Y] = Var[X] + Var[Y] + 2(E [XY] − E [X] E[Y]). e....
IVI U OT 6.00 p Flag question If a and b are constants, the solution of (aye axy+b)dx+(2ye xy + axy’e x – 1)dy=0 is Select one: a.yle axy + bx-y=c b. eaxy + bx= c O c. ye axy + bx - y = c d. y2e xy + x - y=c e. ye ax + bx-y=C f. ye xy +bx+y=c g. y? (e axy - 1) = 0 h. Y x + bx = y
1. Suppose X and Y are continuous random variables with joint pdf f(x,y) 4(z-xy) if = 0 < x < 1 and 0 < y < 1, and zero otherwise. (a) Find E(XY) b) Find E(X-Y) (c) Find Var(X - Y) (d) What is E(Y)?
Let X and Y be two independent random variables with X =d R(0, 2) and Y =d exp(1). (a) Use the convolution formula to calculate the probability density function of W =X+Y. (b) Derive the probability density function of U = XY .
Problem 5 of 5Sum of random variables Let Mr(μ, σ2) denote the Gaussian (or normal) pdf with Inean ,, and variance σ2, namely, fx (x) = exp ( 2-2 . Let X and Y be two i.i.d. random variables distributed as Gaussian with mean 0 and variance 1. Show that Z-XY is again a Gaussian random variable but with mean 0 and variance 2. Show your full proof with integrals. 2. From above, can you derive what will be the...
Suppose X, Y and Z are three different random variables. Let X obey Bernoulli Distribution. The probability distribution function is p(x) = Let Y obeys the standard Normal (Gaussian) distribution, which can be written as Y ∼ N(0, 1). X and Y are independent. Meanwhile, let Z = XY . (a) What is the Expectation (mean value) of X? (b) Are Y and Z independent? (Just clarify, do not need to prove) (c) Show that Z is also a standard...
For random variables X, Y, and Z, Var(X) = 4, Var(Y) = 9, Var(Z) = 16, E[XY] = 6, E[XZ] = −8, E[Y Z] = 10, E[X] = 1, E[Y ] = 2 and E[Z] = 3. Calculate the followings: (b) Cov(−3Y , −4Z ). (d) Var(Y − 3Z). (e) Var(10X + 5Y − 5Z).