Two random variables are independently distributed if their joint ...

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Two random variables are independently distributed if their joint distribution is the product of their marginal distributions.
 
  It is intuitively easier to understand that two random variables are independently distributed if all conditional distributions of Y given X are equal. Derive one of the two conditions from the other.
  What will be an ideal response?

Question 2

The confidence interval for the sample regression function slope
 
  A) can be used to conduct a test about a hypothesized population regression function slope.
  B) can be used to compare the value of the slope relative to that of the intercept.
  C) adds and subtracts 1.96 from the slope.
  D) allows you to make statements about the economic importance of your estimate.

[hanadaa]

Answer to Question 1

Answer: If all conditional distributions of Y given X are equal, then

But if all conditional distributions are equal, then they must also equal the marginal distribution, i.e.,

Given the definition of the conditional distribution of Y given X = x, you then get

Pr(Y = y = x) = = Pr(Y = y),

which gives you the condition

Pr(Y = y, X = x) = Pr(Y = y) Pr(X = x).

Answer to Question 2

Answer: A

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Gracias!

[Mochi]

Wow, this really help