Question

The molarity of a solute in solution is defined to be the number of moles of solute per liter of solution (1 mole = 6.02 × 1023 molecules). If X is the molarity of a solution of magnesium chloride (MgCl2), and Y is the molarity of a solution of ferric chloride (FeCl3), the molarity of chloride ion (Cl−) in a solution made of equal parts of the solutions of MgCl2 and FeCl3 is given by M = X + 1.5Y . Assume that X has mean 0.125 and standard deviation 0.05, and that Y has mean 0.350 and standard deviation 0.10.

a. Find the mean of M
b. Find the standard deviation of M (assume X and Y are independent).

Please help, need accurate solutions

Answer 1

a) 0.65

b) 0.1581

Explanation:

We are given the following in the question:

Distribution of X:

Mean = 0.125

Standard deviation = 0.05

Distribution of Y:

Mean = 0.350

Standard deviation = 0.10

Solution:

`M = X + 1.5Y`

a) mean of M

`\mu_M = \mu_X + 1.5(\mu_Y) = 0.125 + 1.5(0.350) = 0.65`

b) standard deviation of M

If X and Y are independent then,

`Var(X+Y) = Var(X) + Var(Y)\\Var)aX) = a^2Var(X)`

`Var(M) = Var(X+1.5Y)\\Var(M) =Var(X) + (1.5)^2Var(Y)\\Var(M) = (0.05)^2 + (1.5)^2(0.10)^2 = 0.025`

Standard deviation of M =

`=√(Var(M)) = √(0.025) = 0.1581`