Ashipping company handles containers in three different sizes: (1) 27 ft3 (3 × 3 × 3), (2) 125 ft3, and (3) 512 ft3. let xi (i = 1, 2, 3) denote the number of type i containers shipped during a given week. with μi = e(xi) and σi2 = v(xi), suppose that the mean values and standard deviations are as follows: μ1 = 230 μ2 = 260 μ3 = 150 σ1 = 9 σ2 = 12 σ3 = 7 (a) assuming that x1, x2, x3 are independent, calculate the expected value and variance of the total volume shipped. [hint: volume = 27x1 + 125x2 + 512x3.]

Step-by-step explanation:

Let's start writing the random variables :

X1 : ''The number of 27 containers shipped during a given week''

X2 : ''The number of 125 containers shipped during a given week''

X3 : ''The number of 512 containers shipped during a given week''

We assume that X1, X2 and X3 are independent (data from the exercise).

The random variable T : ''Total volume shipped during a given week'' has the following equation :

We want to calculate E(T) and (σ^2)(T).

In the following equations : (σ^2)(X) = V(X)

The expected value operator works as a linear operator.

Then, we calculate E(T) as the following :

We use the information from the exercise ⇒

For the variance :

If X1, X2, X3, ... , Xn are independent random variables, then ⇒

Applying this to the exercise :

We square the standard deviations to obtain the variance of X1,X2 and X3

they will manufacture 8496 cars