One of Umbrella Corporations manufacturing plants makes two

One of Umbrella Corporation\'s manufacturing plants makes two independent products, X and Y. The cost to make X is $ 0.13 per unit and the cost to make Y is $ 0.56 per unit. There is a monthly fixed cost of $ 5,593 at the plant. Monthly demand for X and Y is random, but it is known that the expected demand for X, E(X), is 1,490 and the expected demand for Y, E(Y), is 2,369.

If C is the total monthly cost to make X and Y, find E(C), to two decimal places. Hint: use the rules of expected value.

Solution

Let the mean number of demand for X and Y in a month be \'X\' and \'Y\' respectively.

So, the total exepected cost will be -

C = $(0.13*X + 0.56*Y)

So, E[C] = E[0.13*X + 0.56*Y]

Now, from the property of expected values, we know that E[a + b] = E[a] + E[b].

So, E[C] = E[0.13*X] + E[0.56*Y]

Also, the rule of expected vaalue says that E[cX] = c*E[X], where \'c\' is a constant.

So, E[C] = 0.13*E[X] + 0.56*E[Y]

And it is given that E[X] = 1,490 and E[Y] = 2,369.

So, E[C] = $[0.13*(1490) + 0.56*(2369)]

=> E[C] = $[193.7 + 1326.64]

=> E[C] = $1520.34

Hence, E[C] = $1520.34