A chemical supply company currenbly has in stock 100 lb of a certain chemical, w

Question

A chemical supply company currenbly has in stock 100 lb of a certain chemical, which it sells to customers in 6-lb batches, Let x a the number of batches ordered by a randomly chosen customer, and suppose that X has the following pmf. p) 02 040.3 Compute E(X) and VO). batches compute the expected expected welght let variance of weight left nber of pounds left after tho "est ontoner's order is shipped and the wariance of the mber of pounda left Aent. The umber of pounds left is a sne-function or x.

Explanation / Answer

a) The expected value of X is computed as:

E(X) = 1*0.2 + 2*0.4 + 3*0.3 + 4*0.1 = 0.2 + 0.8 + 0.9 + 0.4 = 2.3

Therefore E(X) = 2.3

The second moment is computed as:

E(X2) = 12*0.2 + 22*0.4 + 32*0.3 + 42*0.1 = 0.2 + 1.6 + 2.7 + 1.6 = 6.1

The variance is now computed as:

V(X) = E(X2) - (E(X))2 = 6.1 - 2.32 = 0.81

Therefore V(X) = 0.81

Now the remaining amount can be modelled as a random variable Y such that

Y = 100 - 6X because each batch has 6 lbs.

Therefore,

E(Y) = E(100 - 6X) = 100 - 6E(X) = 100 - 6*2.3 = 86.2

Therefore E(Y) = 86.2

Now similarly,

Var(Y) = Var(100 - 6X) = 62Var(X) = 36*0.81 = 29.16

Note that the constant can be taken out of the variance operator only after squaring it.

Therefore V(Y) = 29.16

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