## Use part (a) to show that X and Y are independent.

1.  Refer to Exercise 30. An equation to predict the ductility of a titanium weld is Y = 7.84C + 11.44N + O − 1.58Fe, where Y is the oxygen equivalence used to predict ductility, and C, N, O, and Fe are the amounts of carbon, nitrogen, oxygen, and iron, respectively, in weight percent, in the weld. Using the means, standard deviations, and correlations presented in Exercise 30, find µY and σY

2.  Let X and Y be jointly continuous with joint probability density function f(x, y) and marginal densities fX (x) and fY (y). Suppose that f(x, y) = g(x)h(y) where g(x) is a function of x alone, h(y) is a function of y alone, and both g(x) and h(y) are nonnegative.

a.  Show that there exists a positive constant c such that fX (x) = cg(x) and fY (y) = (1/c)h(y).

b.   Use part (a) to show that X and Y are independent.

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

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