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# Linear Statistical Models - True or False

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Question 2

Decide if the following statements are true or false. Justify your answers.

a) If X and Y are two random variables with zero coefﬁcient of correlation then X and Y are independent.

b) If X and Y are independent random variables, then their coeﬂicient of correlation is zero.

c) Let X and Y be two random variables with zero coefficient of correlation. Then there is no linear association between X and Y.

d) If the correlation coefficient between X and Y is positive, then as X increases, Y tends to decrease.

e) See attached

f) See attached

g) In the simple linear regression model there are two parameters to be estimated.

h) See attached

i) In the simple linear regression model: see attached

j) If in a simple linear regression model r = 0.25 then 25% of the variation of y is explained by x.

k) In the simple linear regression model (see attached), we have (see attached).

##### Solution Summary

The solution decides if the given statements on linear statistical models are true or false.

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False.
The Pearson coefficient of correlation between two variables X,Y is defined as
(1.1)
Let be a random variable with a symmetric distribution function P(x) about x=0 (say normal distribution) with zero mean:
(1.2)
Define
(1.3)
Obviously X and Y are NOT independent, however, due to the symmetry of the distribution function we have an odd integrand when we calculate :
(1.4)
So the correlation is:
(1.5)
And we see that even if the correlation between X and Y is zero, it does not mean they are independent.
2.

True
Independent ...

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