Homework Answers
2.)
a) The expected value of a constant is just the constant.
b) If X and Y are the two random variables, then the mathematical expectation of the sum of the two variables is equal to the sum of the mathematical expectation of X and the mathematical expectation of Y, provided that the mathematical expectation exists.
c) if X and Y are the two random variables, then the mathematical expectation of the division of the two variables is not equal to the mathematical expectation of X divided by mathematical expectation of Y, provided that the mathematical expectation exists.
d) The mathematical expectation of the product of the two random variables will not be the product of the mathematical expectation of those two variables.
Note that,
the mathematical expectation of the product of the two random variables will be the product of the mathematical expectation of those two variables, provided that the two variables are independent in nature. In other words, E(XY)=E(X)E(Y).
e) The expected value of square of random variable X is not be tha square of expected value of X.
f) the mathematical expectation of the product of a constant and the function of a random variable is equal to the product of the constant and the mathematical expectation of the function of that random variable provided that their mathematical expectation exists.
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2. Explain in words, and words only, the following properties of expected values. NOTE: X and...
2. Explain in words, and words only, the following properties of expected values. NOTE: X and Y are random variab...
2. Explain in words, and words only, the following properties of expected values. NOTE: X and Y are random variables and k is a constant. (a) E(k) = k (b) E(X+Y) = E(X) + E(Y) (c) E(X/Y) + E(X)/E(Y) (d) E(X+Y) E(X)*E(Y) (unless what?) (e) E(X2) # (E(X)]? (1) E(kX) = E(X) 3. For random variable X with mean H. variance is defined var(X) = Ef(X-M.)'. Show how variance can be expressed only in terms of E(X) and E(X). 4....
2. Properties of Correlation and Covariance: Two random variables Y and Z are represented by the...
2. Properties of Correlation and Covariance: Two random variables Y and Z are represented by the following relationships Y = 0.5+0.6X Z = 0.2+0.3x where X is another random variable. You can treat the variance, Var(X), as a given constant. It may help to give Var(X) a name, ie. Var(x)ox2 a. Calcuate var(Y) and Var(Z) as a function of Var(X). Which is hrger? b. Calcuate Cov(Y,Z), Cov(X,Z) and Cov(X,Y) as a function of var(X). c. Calcuate Corr(Y,Z), Corr(X,Z) and Corn(X,Y)...
Prove the following properties using the definition of the variance and the covariance: Q1. Operations with...
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,...
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....
In a similar manner, prove the following properties using the definition of the variance and the...
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)
14. Random variables X and Y have a density function f(x, y). Find the indicated expected...
14. Random variables X and Y have a density function f(x, y). Find the indicated expected value. f(x, y) = (xy + y2) 0<x< 1,0 <y<1 0 Elsewhere {$(wyty E(x2y) = 15. The means, standard deviations, and covariance for random variables X, Y. and Z are given below. LIX = 3. HY = 5. Az = 7 Ox= 1, = 3, oz = 4 cov(X,Y) = 1, cov (X, Z) = 3, and cov (Y,Z) = -3 T = X-2...
Q2. More about operations with expectation and covariances Recall that the variance of random variable X...
Q2. More about operations with expectation and covariances Recall that the variance of random variable X is defined as Var(X) Ξ E 1(X-E(X))2」, the covariance is Cor(X, Y-E (X-E(X))(Y-E(Y)), and the correlation is Corr(X,Y) Ξ (a) What is the value of EX-E(X))? (Hint: Let μ denote E(X). Then, the parameter μ is a unknown, but fixed value like a constant.) (0.5 pt) b) The following is the proof that Var(X) E(X2) E(X)2: -E(x)-E(x)2 In a similar way, prove that Cov(X,...
Recall that the variance of a random variable is defined as Var[X]=E[(X−μ)2], where μ = E[X]....
Recall that the variance of a random variable is defined as
Var[X]=E[(X−μ)2], where μ = E[X]. Use the properties of
expectation to show that we can rewrite the variance of a random
variable X as Var [X]=E[X^2]−(E[X])^2
Problem 3. (1 point) Recall that the variance of a random variable is defined as Var X-E(X-μ)21, where μ= E[X]. Use the properties of expectation to show that we can rewrite the variance of a random variable X as u hare i- ElX)L...
5.8.6 otherwise. (a) Find the correlation rx.y (b) Find the covariance Cov(X,Y]. 5.8.6 The random variables...
5.8.6
otherwise. (a) Find the correlation rx.y (b) Find the covariance Cov(X,Y]. 5.8.6 The random variables X and Y have (b) Use part Cov oint PMF (c) Show tha Var[ (d) Combine Px,y and 5.8.10 Ran the joint PM PN,K (n, k) 0 0 Find (a) The expected values E[X] and EY, pected (b) The variances Var(X] and Var[Y],VarlK], E Find the m
Let X, Y, Z be random variables with these properties: · E[X] = 3 and E[X²]...
Let X, Y, Z be random variables with these properties: · E[X] = 3 and E[X²] = 10 Var(Y) = 5 E[Z] = 2 and E[Z2] = 7 • X and Y are independent E[X2] = 5 Cov(Y,Z) = 2 Find Var(3X+Y – Z).
2. Explain in words, and words only, the following properties of expected values. NOTE: X and Y are random variables and k is a constant. (a) E(k) = k (b) E(X+Y) = E(X) + E(Y) (c) E(X/Y) + E(X)/E(Y) (d) E(X+Y) E(X)*E(Y) (unless what?) (e) E(X2) # (E(X)]? (1) E(kX) = E(X) 3. For random variable X with mean H. variance is defined var(X) = Ef(X-M.)'. Show how variance can be expressed only in terms of E(X) and E(X). 4....
2. Properties of Correlation and Covariance: Two random variables Y and Z are represented by the following relationships Y = 0.5+0.6X Z = 0.2+0.3x where X is another random variable. You can treat the variance, Var(X), as a given constant. It may help to give Var(X) a name, ie. Var(x)ox2 a. Calcuate var(Y) and Var(Z) as a function of Var(X). Which is hrger? b. Calcuate Cov(Y,Z), Cov(X,Z) and Cov(X,Y) as a function of var(X). c. Calcuate Corr(Y,Z), Corr(X,Z) and Corn(X,Y)...
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,...
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....
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)
14. Random variables X and Y have a density function f(x, y). Find the indicated expected value. f(x, y) = (xy + y2) 0<x< 1,0 <y<1 0 Elsewhere {$(wyty E(x2y) = 15. The means, standard deviations, and covariance for random variables X, Y. and Z are given below. LIX = 3. HY = 5. Az = 7 Ox= 1, = 3, oz = 4 cov(X,Y) = 1, cov (X, Z) = 3, and cov (Y,Z) = -3 T = X-2...
Q2. More about operations with expectation and covariances Recall that the variance of random variable X is defined as Var(X) Ξ E 1(X-E(X))2」, the covariance is Cor(X, Y-E (X-E(X))(Y-E(Y)), and the correlation is Corr(X,Y) Ξ (a) What is the value of EX-E(X))? (Hint: Let μ denote E(X). Then, the parameter μ is a unknown, but fixed value like a constant.) (0.5 pt) b) The following is the proof that Var(X) E(X2) E(X)2: -E(x)-E(x)2 In a similar way, prove that Cov(X,...
Recall that the variance of a random variable is defined as
Var[X]=E[(X−μ)2], where μ = E[X]. Use the properties of
expectation to show that we can rewrite the variance of a random
variable X as Var [X]=E[X^2]−(E[X])^2
Problem 3. (1 point) Recall that the variance of a random variable is defined as Var X-E(X-μ)21, where μ= E[X]. Use the properties of expectation to show that we can rewrite the variance of a random variable X as u hare i- ElX)L...
5.8.6
otherwise. (a) Find the correlation rx.y (b) Find the covariance Cov(X,Y]. 5.8.6 The random variables X and Y have (b) Use part Cov oint PMF (c) Show tha Var[ (d) Combine Px,y and 5.8.10 Ran the joint PM PN,K (n, k) 0 0 Find (a) The expected values E[X] and EY, pected (b) The variances Var(X] and Var[Y],VarlK], E Find the m
Let X, Y, Z be random variables with these properties: · E[X] = 3 and E[X²] = 10 Var(Y) = 5 E[Z] = 2 and E[Z2] = 7 • X and Y are independent E[X2] = 5 Cov(Y,Z) = 2 Find Var(3X+Y – Z).
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