a.Is this statement true: there exist two independent random variables X and Y s
ID: 3292808 • Letter: A
Question
a.Is this statement true: there exist two independent random variables X and Y such that Var [X] = Var [Y] = 1? If true, find such example, otherwise prove why this is impossible.
b. Consider this question related to a. Is this statement true: there exist two independent random variables X and Y such that Var [X] = Var [Y] = 1, and also E[X] = E[Y] = 0? If true, find such example, otherwise prove why this is impossible.
c. Consider this question related to a and b. . Is this statement true: there exist two independent random variables X and Y such that Var [X] = Var [Y] = 1;E[X] = E[Y] = 0, and also Cov [X, Y] = 0.5? If true, find such example, otherwise prove why this is impossible. (here Cov [X, Y] stands for the covariance of X and Y.
d. Consider this question related to a, b, and c . Is this statement true: there exist two independent random variables X and Y such that Var [X] = Var [Y] = 1;E[X] = E[Y] = 0, and also Cov [X, Y] = 0? If true, find such example, otherwise prove why this is impossible.
Explanation / Answer
a) True
Example : X~N(0,1) and Y~N(0,1) where X and Y are independent random variables.
b) True
Example : X~N(0,1) and Y~N(0,1) where X and Y are independent random variables.
c) Covariance between two random variables is a measure of linear relation between X and Y.
If X and Y are independent random variables, then implies there is no relationship between them that implies Cov(X,Y) should be equal to 0, it can not be 0.5
d) True
Example : X and Y are two independent standard normal variables.
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