Answer:
The expected repair cost is $3.73.
Step-by-step explanation:
The random variable <em>X</em> is defined as the number of defectives among the 4 items sold.
The probability of a large lot of items containing defectives is, <em>p</em> = 0.09.
An item is defective irrespective of the others.
The random variable <em>X</em> follows a Binomial distribution with parameters <em>n</em> = 9 and <em>p</em> = 0.09.
The repair cost of the item is given by:
![C=3X^{2}+X+2](https://tex.z-dn.net/?f=C%3D3X%5E%7B2%7D%2BX%2B2)
Compute the expected cost of repair as follows:
![E(C)=E(3X^{2}+X+2)](https://tex.z-dn.net/?f=E%28C%29%3DE%283X%5E%7B2%7D%2BX%2B2%29)
![=3E(X^{2})+E(X)+2](https://tex.z-dn.net/?f=%3D3E%28X%5E%7B2%7D%29%2BE%28X%29%2B2)
Compute the expected value of <em>X</em> as follows:
![E(X)=np](https://tex.z-dn.net/?f=E%28X%29%3Dnp)
![=4\times 0.09\\=0.36](https://tex.z-dn.net/?f=%3D4%5Ctimes%200.09%5C%5C%3D0.36)
The expected value of <em>X</em> is 0.36.
Compute the variance of <em>X</em> as follows:
![V(X)=np(1-p)](https://tex.z-dn.net/?f=V%28X%29%3Dnp%281-p%29)
![=4\times 0.09\times 0.91\\=0.3276\\](https://tex.z-dn.net/?f=%3D4%5Ctimes%200.09%5Ctimes%200.91%5C%5C%3D0.3276%5C%5C)
The variance of <em>X</em> is 0.3276.
The variance can also be computed using the formula:
![V(X)=E(Y^{2})-(E(Y))^{2}](https://tex.z-dn.net/?f=V%28X%29%3DE%28Y%5E%7B2%7D%29-%28E%28Y%29%29%5E%7B2%7D)
Then the formula of
is:
![E(Y^{2})=V(X)+(E(Y))^{2}](https://tex.z-dn.net/?f=E%28Y%5E%7B2%7D%29%3DV%28X%29%2B%28E%28Y%29%29%5E%7B2%7D)
Compute the value of
as follows:
![E(Y^{2})=V(X)+(E(Y))^{2}](https://tex.z-dn.net/?f=E%28Y%5E%7B2%7D%29%3DV%28X%29%2B%28E%28Y%29%29%5E%7B2%7D)
![=0.3276+(0.36)^{2}\\=0.4572](https://tex.z-dn.net/?f=%3D0.3276%2B%280.36%29%5E%7B2%7D%5C%5C%3D0.4572)
The expected repair cost is:
![E(C)=3E(X^{2})+E(X)+2](https://tex.z-dn.net/?f=E%28C%29%3D3E%28X%5E%7B2%7D%29%2BE%28X%29%2B2)
![=(3\times 0.4572)+0.36+2\\=3.7316\\\approx 3.73](https://tex.z-dn.net/?f=%3D%283%5Ctimes%200.4572%29%2B0.36%2B2%5C%5C%3D3.7316%5C%5C%5Capprox%203.73)
Thus, the expected repair cost is $3.73.