Homework Answers
Answer:
a).
when x increases, y decreases. The relation is negative. The covariance is a negative value.
Possible value may be around -2
Note:
approximate calculation for covariance.
Assume correlation is about -0.9 and by range rule, sy=range/4=1.25 and sx=range/4=1.75
Covariance = r*sx*sy = -0.9*1.25*1.75 = -1.968
b). X and Y are not independent because clearly when x increases y decreases.
When X=10 we expect Y value is around 1.
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pr 1 X and Y are two random variables. A scatter diagram for a 15 different...
X and Y are two random variables. A scatter diagram for a 15 differest samples of...
X and Y are two random variables. A scatter diagram for a 15 differest samples of both tandom yariables is illustrated below 7 t. pt. a. | Suggest a convenient value for the covarance of X, Y . Is X. Y independeut? Why? If they are dependent and you know that X-10, what2 pt 2 pt. you conld expect about Y GOOD LUCK
9. Let X and Y be Bernouilli random variables with joint distribution: Pr(X = 1 and...
9. Let X and Y be Bernouilli random variables with joint distribution: Pr(X = 1 and Y = 1) = 0.42, Pr(X = 1 and Y = 0) = 0.18, Pr(X = 0 and Y = 1) = 0.28, Pr(X = 0 and Y = 0) = 0.12 Determine whether or not X and Y are independent. Determine the covariance and correlation for X and Y. Can you infer anything from this correlation?
2. The covariance of two random variables is Oxy = (x-7)(y-7) Show that the covariance is...
2. The covariance of two random variables is Oxy = (x-7)(y-7) Show that the covariance is zero if the random variables are independent.
4.2 The Correlation Coefficient 1. Let the random variables X and Y have the joint PMF...
4.2 The Correlation Coefficient 1. Let the random variables X and Y have the joint PMF of the form x + y , x= 1,2, y = 1,2,3. p(x,y) = 21 They satisfy 11 12 Mx = 16 of = 12 of = 212 2 My = 27 Find the covariance Cov(X,Y) and the correlation coefficient p. Are X and Y independent or dependent?
1. Consider a pair of random variables (X, Y) with joint PDF fx,y(x, y) 0, otherwise....
1. Consider a pair of random variables (X, Y) with joint PDF fx,y(x, y) 0, otherwise. (a) 1 pt - Find the marginal PDF of X and the marginal PDF of Y. (b) 0.5 pt - Are X and Y independent? Why? (e) 0.5 pt - Compute the mean of X and the mean of Y.
Let X and Y be two independent and identically distributed random variables with expected value 1...
Let X and Y be two independent and identically distributed random variables with expected value 1 and variance 2.56. First, find a non-trivial upper bound for P(|X + Y − 2| ≥ 1). Now suppose that X and Y are independent and identically distributed N(1,2.56) random variables. What is P(|X + Y − 2| ≥ 1) exactly? Why is the upper bound first obtained so different from the exact probability obtained?
2) Two statistically-independent random variables, (X,Y), each have marginal probability density,...
2) Two statistically-independent random variables, (X,Y), each have marginal probability density, N(0,1) (e.g., zero-mean, unit-variance Gaussian). Let V-3X-Y, Z = X-Y Find the covariance matrix of the vector,
2) Two statistically-independent random variables, (X,Y), each have marginal probability density, N(0,1) (e.g., zero-mean, unit-variance Gaussian). Let V-3X-Y, Z = X-Y Find the covariance matrix of the vector,
1. Suppose X,Y are random variables whose joint pdf is given by f(x, y) = 1/...
1. Suppose X,Y are random variables whose joint pdf is given by f(x, y) = 1/ x , if 0 < x < 1, 0 < y < x f(x, y) =0, otherwise . Find the covariance of the random variables X and Y . 2.Let X1 be a Bernoulli random variable with parameter p1 and X2 be a Bernoulli random variable with parameter p2. Assume X1 and X2 are independent. What is the variance of the random variable Y...
1. Suppose you have two random variables, X and Y with joint distribution given by the...
1. Suppose you have two random variables, X and Y with joint distribution given by the following tables So, for example, the probability that Y o,x - 0 is 4, and the probability that Y (a) Find the marginal distributions (pmfs) of X and Y, denoted f(x),J(Y). (b) Find the conditional distribution (pmf) of Y give X, denoted f(YX). (c) Find the expected values of X and Y, EX), E(Y). (d) Find the variances of X and Y, Var(X),Var(Y). (e)...
Suppose the random variables X, Y and Z are related through the model Y = 2...
Suppose the random variables X, Y and Z are related through the
model
Y = 2 + 2X + Z,
where Z has mean 0 and variance σ2 Z = 16 and X has variance σ2
X = 9. Assume X and Z are independent, the find the covariance of X
and Y and that of Y and Z. Hint: write Cov(X, Y ) = Cov(X, 2+2X+Z)
and use the propositions of covariance from slides of Chapter
4.
Suppose the...
X and Y are two random variables. A scatter diagram for a 15 differest samples of both tandom yariables is illustrated below 7 t. pt. a. | Suggest a convenient value for the covarance of X, Y . Is X. Y independeut? Why? If they are dependent and you know that X-10, what2 pt 2 pt. you conld expect about Y GOOD LUCK
9. Let X and Y be Bernouilli random variables with joint distribution: Pr(X = 1 and Y = 1) = 0.42, Pr(X = 1 and Y = 0) = 0.18, Pr(X = 0 and Y = 1) = 0.28, Pr(X = 0 and Y = 0) = 0.12 Determine whether or not X and Y are independent. Determine the covariance and correlation for X and Y. Can you infer anything from this correlation?
2. The covariance of two random variables is Oxy = (x-7)(y-7) Show that the covariance is zero if the random variables are independent.
4.2 The Correlation Coefficient 1. Let the random variables X and Y have the joint PMF of the form x + y , x= 1,2, y = 1,2,3. p(x,y) = 21 They satisfy 11 12 Mx = 16 of = 12 of = 212 2 My = 27 Find the covariance Cov(X,Y) and the correlation coefficient p. Are X and Y independent or dependent?
1. Consider a pair of random variables (X, Y) with joint PDF fx,y(x, y) 0, otherwise. (a) 1 pt - Find the marginal PDF of X and the marginal PDF of Y. (b) 0.5 pt - Are X and Y independent? Why? (e) 0.5 pt - Compute the mean of X and the mean of Y.
2) Two statistically-independent random variables, (X,Y), each have marginal probability density, N(0,1) (e.g., zero-mean, unit-variance Gaussian). Let V-3X-Y, Z = X-Y Find the covariance matrix of the vector,
2) Two statistically-independent random variables, (X,Y), each have marginal probability density, N(0,1) (e.g., zero-mean, unit-variance Gaussian). Let V-3X-Y, Z = X-Y Find the covariance matrix of the vector,
1. Suppose you have two random variables, X and Y with joint distribution given by the following tables So, for example, the probability that Y o,x - 0 is 4, and the probability that Y (a) Find the marginal distributions (pmfs) of X and Y, denoted f(x),J(Y). (b) Find the conditional distribution (pmf) of Y give X, denoted f(YX). (c) Find the expected values of X and Y, EX), E(Y). (d) Find the variances of X and Y, Var(X),Var(Y). (e)...
Suppose the random variables X, Y and Z are related through the
model
Y = 2 + 2X + Z,
where Z has mean 0 and variance σ2 Z = 16 and X has variance σ2
X = 9. Assume X and Z are independent, the find the covariance of X
and Y and that of Y and Z. Hint: write Cov(X, Y ) = Cov(X, 2+2X+Z)
and use the propositions of covariance from slides of Chapter
4.
Suppose the...
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