Derive The Following Formulas
(a) P(Ac)=1−P(A).
(b) P(A∪B)=P(A)+P(B)−P(A∩B)
(c) P(A ∩ B) = P(A|B)P(B)
(d) E(aX + b) = aE(X) + b where you will be told whether X is assumed to be discrete or
X is assumed to be continuous.
(e) Var(X) = E(X2) − μ2 where you will be told whether X is assumed to be discrete or X
is assumed to be continuous.
(f) Var(aX + b) = a2Var(X)
Derive The Following Formulas (a) P(Ac)=1−P(A). (b) P(A∪B)=P(A)+P(B)−P(A∩B) (c) P(A ∩ B) = P(A|B)P(B) (d) E(aX...
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....
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)
Problem 1 Suppose X ~ fx(x), and let Y = aX + b. We know that E(Y) = aE(X) b, and Var(X)a2Var(X). What about the density of Y, fy(y)? Assuming a > 0. Calculate fy(y) using the following two methods (1) Let Fx() P(X x). Calculate Fy(y) = P(Y < y) in terms of Fx. Then calculate fy (2) Calculate Y (y, y + Ay)) Ay fr(y) (3) Give geometric explanations of your result
Prove the following properties using the definition of the
variance and the covariance:
Q1. Operations with expectation and covariances Recall that the variance of randon variable X is defined as Var(X) Ξ E [X-E(X))2], the covariance is Cov(X, ) EX E(X))Y EY) As a hint, we can prove Cov(aX + b, cY)-ac Cov(X, Y) by ac EX -E(X)HY -E(Y)ac Cov(X, Y) In a similar manner, prove the following properties using the definition of the variance and the covariance: (a) Var(X)-Cov(X,...
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....
1. A binomial random variable has the moment generating function, (t) E(etx)II1 E(etX) (pet+1-p)". Show that EX] = np and Var(X) = np(1-p) using that EX] = ψ(0) and E(X2] = ψ"(0). 2. Lex X be uniformly distributed over (a,b). Show that E[xt and Var(X) using the first and second moments of this random variable where the pdf of X is f(x). Note that the nth moment of a continuous random variable is defined as EXj-Γοχ"f(x)dx (b-a)2 exp 2
Let X and Y b Var(Y) (1) If a, b,c and d are fixed real numbers, = E(X), μγ E (Y),咳= Var(X) and e ranclom variables. with y a) show Cov(aX +b, cY +d)- ac Cov(X,Y) (b) show Corr(aX + b, cY + d)-PXY for a > 0 and c > 0.
Derive the propagated errors SG or SGul for the following formulas (x, y, z denote variables and a, b constants) a • G(x) = ax b • G(x) = 1/2 c • G(x) = 1/(x + a) d • G(x) = a log(bx) e • (x) = ax! • G(x) = a exp(-bx) g • G(x, y, z) = xyz h • G(x, y) = xy
(D) Rules of Expectations and Variances. Obtain the following expectations and variances to the extent possible. For simplicity assume that X and Y are both continuous (a) E(Y) and Var(Y) where Y-3x +0.5 (Note that this definition of Y only holds for this part.) (b) E(ax bY) where a and b are positive constants. (c) E(X·Y) assuming that X and Y are independent (d) Var(aX +bY) where a and b are positive constants and X and Y are independent.
In a similar manner, prove the following properties using the definition of the variance and the covariance: (a) Var(X) - Cov(X, X). (0.5 pt) (b) Cov(X,a)-0. (0.5 pt) (c) Cov(aX, Y)aCov(X, Y) (0.5 pt) (d) Cov(aX,bY) -abCov(X, Y) (0.5 pt) (e) Var(aX) a2Var(X). (0.5 pt)