Answer:
![E(T)=115510ft^{3}](https://tex.z-dn.net/?f=E%28T%29%3D115510ft%5E%7B3%7D)
![V(T)=15154105ft^{6}](https://tex.z-dn.net/?f=V%28T%29%3D15154105ft%5E%7B6%7D)
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 :
![T=(27ft^{3})X_{1}+(125ft^{3})X_{2}+(512ft^{3})X_{3}](https://tex.z-dn.net/?f=T%3D%2827ft%5E%7B3%7D%29X_%7B1%7D%2B%28125ft%5E%7B3%7D%29X_%7B2%7D%2B%28512ft%5E%7B3%7D%29X_%7B3%7D)
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 :
![E(T)=E(27X1+125X2+512X3)](https://tex.z-dn.net/?f=E%28T%29%3DE%2827X1%2B125X2%2B512X3%29)
![E(T)=E(27X1)+E(125X2)+E(512X3)](https://tex.z-dn.net/?f=E%28T%29%3DE%2827X1%29%2BE%28125X2%29%2BE%28512X3%29)
![E(T)=27E(X1)+125E(X2)+512E(X3)](https://tex.z-dn.net/?f=E%28T%29%3D27E%28X1%29%2B125E%28X2%29%2B512E%28X3%29)
We use the information from the exercise ⇒
![E(T)=27.(230)+125.(260)+512.(150)=6210+32500+76800=115510](https://tex.z-dn.net/?f=E%28T%29%3D27.%28230%29%2B125.%28260%29%2B512.%28150%29%3D6210%2B32500%2B76800%3D115510)
![E(T)=115510ft^{3}](https://tex.z-dn.net/?f=E%28T%29%3D115510ft%5E%7B3%7D)
For the variance :
If X1, X2, X3, ... , Xn are independent random variables, then ⇒
![V(a_{1}X_{1}+a_{2}X_{2}+...+a_{n}X_{n})=a_{1} ^{2}V(X_{1})+a_{2} ^{2}V(X_{2})+...+a_{n} ^{2}V(X_{n})](https://tex.z-dn.net/?f=V%28a_%7B1%7DX_%7B1%7D%2Ba_%7B2%7DX_%7B2%7D%2B...%2Ba_%7Bn%7DX_%7Bn%7D%29%3Da_%7B1%7D%20%5E%7B2%7DV%28X_%7B1%7D%29%2Ba_%7B2%7D%20%5E%7B2%7DV%28X_%7B2%7D%29%2B...%2Ba_%7Bn%7D%20%5E%7B2%7DV%28X_%7Bn%7D%29)
Applying this to the exercise :
![V(T)=V(27X_{1}+125X_{2}+512X_{3})\\](https://tex.z-dn.net/?f=V%28T%29%3DV%2827X_%7B1%7D%2B125X_%7B2%7D%2B512X_%7B3%7D%29%5C%5C)
![V(T)=27^{2}V(X_{1})+125^{2}V(X_{2})+512^{2}V(X_{3})](https://tex.z-dn.net/?f=V%28T%29%3D27%5E%7B2%7DV%28X_%7B1%7D%29%2B125%5E%7B2%7DV%28X_%7B2%7D%29%2B512%5E%7B2%7DV%28X_%7B3%7D%29)
We square the standard deviations to obtain the variance of X1,X2 and X3
![V(T)=(27^{2})(9^{2})+(125^{2})(12^{2})+(512^{2})(7^{2})=59049+2250000+12845056](https://tex.z-dn.net/?f=V%28T%29%3D%2827%5E%7B2%7D%29%289%5E%7B2%7D%29%2B%28125%5E%7B2%7D%29%2812%5E%7B2%7D%29%2B%28512%5E%7B2%7D%29%287%5E%7B2%7D%29%3D59049%2B2250000%2B12845056)
![V(T)=15154105(ft^{3})^{2}=15154105ft^{6}](https://tex.z-dn.net/?f=V%28T%29%3D15154105%28ft%5E%7B3%7D%29%5E%7B2%7D%3D15154105ft%5E%7B6%7D)