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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

Respuesta :

Answer:

a) 0.65

b) 0.1581

Step-by-step 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:

[tex]M = X + 1.5Y[/tex]

a) mean of M

[tex]\mu_M = \mu_X + 1.5(\mu_Y) = 0.125 + 1.5(0.350) = 0.65[/tex]

b) standard deviation of M

If X and Y are independent then,

[tex]Var(X+Y) = Var(X) + Var(Y)\\Var)aX) = a^2Var(X)[/tex]

[tex]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[/tex]

Standard deviation of M =

[tex]=\sqrt{Var(M)} = \sqrt{0.025} = 0.1581[/tex]

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