Question

Let X and Y be continuous random variables with joint density function f(x,y) and marginal density functions fX(x) and fY(y) respectively. Further, the support for both of these marginal density functions is the interval (0,1).

Which of the following statements is always true? (Note there may be more than one)

  

E[X^2Y^3]=(∫0 TO 1 x^2 dx)(∫0 TO 1 y^3dy)

  

E[X^2Y^3]=∫0 TO 1∫0 TO 1x^2y^3 f(x,y) dy dx

  

E[Y^3]=∫0 TO 1 y^3 fX(x) dx  

E[XY]=(∫0 TO 1 x fX(x) dx) (∫0 TO 1 y fY(y) dx)

  

E[X^2]=∫0 TO 1 x^2 fX(x) dx

  

E[X2]=∫01x2f(x,y)dxE[X2]=∫01x2f(x,y)dx

  

  

  

  

Answer 1

The correct answers are

• E[X^2Y^3]=∫0 TO 1∫0 TO 1x^2y^3 f(x,y) dy dx
• E[X^2]=∫0 TO 1 x^2 fX(x) dx

Incorrect ansers are :

• E[X^2Y^3]=(∫0 TO 1 x^2 dx)(∫0 TO 1 y^3dy) : Cause: Expectation must contain probability density function
• E[Y^3]=∫0 TO 1 y^3 fX(x) dx , Cause: Expectation must calculated on fY(y) but here it is fX(x)
• E[XY]=(∫0 TO 1 x fX(x) dx) (∫0 TO 1 y fY(y) dx) Cause: This will be correct if X and Y are independent , but in question it is not specified so it is wrong
• E[X2]=∫01x2f(x,y) , Cause : There must be fX(x) in place of f(x,y) to be correct