Prove that Cov(aX, bY) = ab Cov(X,Y).

1.  Let X and Y be random variables, and a and b be constants.

a.  Prove that Cov(aX, bY) = ab Cov(X,Y).

b.  Prove that if a > 0and b > 0, then ρaX,bY = ρX,Y .

c.   Conclude that the correlation coefficient is unaffected by changes in units.


2.   Let X, Y, and Z be jointly distributed random variables. Prove that Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z). (Hint: Use Equation 2.71.)



3.  Let X and Y be jointly distributed random variables. This exercise leads you through a proof of the fact that −1 ≤ ρX,Y ≤ 1.

a.   Express the quantity V(X − (σX /σY )Y) in terms of σx , σY , and Cov(X, Y).

b.  Use the fact that V(X − (σX /σY )Y) ≥ 0 and Cov(X, Y) = ρX,YσXσY to show that ρX,Y ≤ 1.

c.    Repeat parts (a) and (b) using V(X + (σX /σY )Y) to show that ρX,Y ≥ −1.

find the cost of your paper

Determine the outlet concentration x1 in the water solution and in the trichloroethane solution using an analytical equation.

Predicting Extraction for an Existing Tower with a Given Number of Steps. An existing tower contains 5.0 theoretical steps. It is desired to predict its performance under the following conditions….

Calculate the minimum solvent that can be used. [Hint: In this case, the tie line through the feed L0 represents the condition for minimum solvent flow rate.

Minimum Solvent and Countercurrent Extraction of Acetone. An aqueous feed solution of 1000 kg/h containing 23.5 wt % acetone and 76.5 wt % water is being extracted in a countercurrent….

Determine the break-point time, the fraction of total capacity used up to the break point, the length of the unused bed, and the saturation loading capacity of the solid.

Drying of Nitrogen and Scale-Up of a Column. Using molecular sieves, water vapor was removed from nitrogen gas in a packed bed (C1) at 28.3°C. The column height was 0.268….