Exercise 2 (2). Let X and ε be independent normally distributed random variables such that X∼N(5,4)andε∼N(0,9).LetY...
Exercise 2 (2). Let X and ε be independent normally distributed
random variables such that X∼N(5,4)andε∼N(0,9).LetY
bearandomvariablegivenbyY =1+2X+ε.Compute: (a) E(Y )
(b) Var(Y )
(c) Cov(X, Y )
(d) Corr(X, Y )
(e) What is the value of the ratio Cov(X, Y )/Var(X) ?
(f) If Y = 1 + 3X + ε instead, what would be the value of Cov(X, Y
)/Var(X) ? (g) If Y = 1 + 7X + ε instead, what would be the value
of Cov(X, Y )/Var(X) ?
Homework Answers
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Exercise 2 (2). Let X and ε be independent normally distributed
random variables such that X∼N(5,4)andε∼N(0,9).LetY...
9. Let X and Y be independent and identically distributed random variables with mean u and...
9. Let X and Y be independent and identically distributed random variables with mean u and variance o. Find the following: (a) E[(x + 2)] (b) Var(3x + 4) (c) E[(X-Y)] (d) Cov{(X + Y), (X - Y)}
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 )
Can someone help me with this? Show that two jointly normally distributed random variables are independent...
Can someone help me with this? Show that two jointly normally
distributed random variables are independent if they are
uncorrelated?
Additional Info:
Thank's a lot!!!
Let (*) ~ ~[(*) (*)) with oš> 0, 0} > 0. NX Then YlX^N (wy +O20yx(– Hx), oz, – 022Oxy@yx). That is, the regression function is here linear (in X): E[Y|X] = E[Y]+B(X – E[X]) = Hy +B(X – Hx), where B = Cov(X, Y) = pºy; recall: =vx= POD Cov(X, Y) = Oxy =...
(Sums of normal random variables) Let X be independent random variables where XN N(2,5) and Y...
(Sums of normal random variables) Let X be independent random variables where XN N(2,5) and Y ~ N(5,9) (we use the notation N (?, ?. ) ). Let W 3X-2Y + 1. (a) Compute E(W) and Var(W) (b) It is known that the sum of independent normal distributions is n Compute P(W 6)
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).
1. Let X,.., Xn be independent and identically distributed as N (0,9) (here 9 is the...
1. Let X,.., Xn be independent and identically distributed as N (0,9) (here 9 is the variane (a) What is the distribution of Y-1X,? (Verify it using MGF) (b) What is the distribution of Xn X? (Again verify it using MGF) (c) Assume n -25. What is the probability that an observed value of X lies inside the interval [-1.2,1.2] (d) Give a lower bound on the probability that Xn lies inside the interval1.2,1.2] using Chebyshev's inequality. Compare it with...
Exercise 1 (1). X, Y are random variables (r.v.) and a,b,c,d are values. Complete the formulas...
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)
Let A, B, and C be independent random variables, uniformly distributed over [0,9], [0,2], and [0,3]...
Let A, B, and C be independent random variables, uniformly distributed over [0,9], [0,2], and [0,3] respectively. What is the probability that both roots of the equation Ax^2+Bx+C=0 are real?
Let X and Y be two independent random variables. Show that Cov (X, XY) = E(Y)...
Let X and Y be two independent random variables. Show that Cov (X, XY) = E(Y) Var(X).
5. Suppose X is a normally distributed random variable with mean μ and variance 2. Consider...
5. Suppose X is a normally distributed random variable with mean μ and variance 2. Consider a new random variable, W=2X + 3. i. What is E(W)? ii. What is Var(W)? 6. Suppose the random variables X and Y are jointly distributed. Define a new random variable, W=2X+3Y. i. What is Var(W)? ii. What is Var(W) if X and Y are independent?
9. Let X and Y be independent and identically distributed random variables with mean u and variance o. Find the following: (a) E[(x + 2)] (b) Var(3x + 4) (c) E[(X-Y)] (d) Cov{(X + Y), (X - Y)}
Can someone help me with this? Show that two jointly normally
distributed random variables are independent if they are
uncorrelated?
Additional Info:
Thank's a lot!!!
Let (*) ~ ~[(*) (*)) with oš> 0, 0} > 0. NX Then YlX^N (wy +O20yx(– Hx), oz, – 022Oxy@yx). That is, the regression function is here linear (in X): E[Y|X] = E[Y]+B(X – E[X]) = Hy +B(X – Hx), where B = Cov(X, Y) = pºy; recall: =vx= POD Cov(X, Y) = Oxy =...
(Sums of normal random variables) Let X be independent random variables where XN N(2,5) and Y ~ N(5,9) (we use the notation N (?, ?. ) ). Let W 3X-2Y + 1. (a) Compute E(W) and Var(W) (b) It is known that the sum of independent normal distributions is n Compute P(W 6)
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).
1. Let X,.., Xn be independent and identically distributed as N (0,9) (here 9 is the variane (a) What is the distribution of Y-1X,? (Verify it using MGF) (b) What is the distribution of Xn X? (Again verify it using MGF) (c) Assume n -25. What is the probability that an observed value of X lies inside the interval [-1.2,1.2] (d) Give a lower bound on the probability that Xn lies inside the interval1.2,1.2] using Chebyshev's inequality. Compare it with...
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)
Let X and Y be two independent random variables. Show that Cov (X, XY) = E(Y) Var(X).
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