Respuesta :
Answer:
The expected repair cost is $3.73.
Step-by-step explanation:
The random variable X is defined as the number of defectives among the 4 items sold.
The probability of a large lot of items containing defectives is, p = 0.09.
An item is defective irrespective of the others.
The random variable X follows a Binomial distribution with parameters n = 9 and p = 0.09.
The repair cost of the item is given by:
[tex]C=3X^{2}+X+2[/tex]
Compute the expected cost of repair as follows:
[tex]E(C)=E(3X^{2}+X+2)[/tex]
    [tex]=3E(X^{2})+E(X)+2[/tex]
Compute the expected value of X as follows:
[tex]E(X)=np[/tex]
     [tex]=4\times 0.09\\=0.36[/tex]
The expected value of X is 0.36.
Compute the variance of X as follows:
[tex]V(X)=np(1-p)[/tex]
     [tex]=4\times 0.09\times 0.91\\=0.3276\\[/tex]
The variance of X is 0.3276.
The variance can also be computed using the formula:
[tex]V(X)=E(Y^{2})-(E(Y))^{2}[/tex]
Then the formula of [tex]E(Y^{2})[/tex] is:
[tex]E(Y^{2})=V(X)+(E(Y))^{2}[/tex]
Compute the value of [tex]E(Y^{2})[/tex] as follows:
[tex]E(Y^{2})=V(X)+(E(Y))^{2}[/tex]
     [tex]=0.3276+(0.36)^{2}\\=0.4572[/tex]
The expected repair cost is:
[tex]E(C)=3E(X^{2})+E(X)+2[/tex]
     [tex]=(3\times 0.4572)+0.36+2\\=3.7316\\\approx 3.73[/tex]
Thus, the expected repair cost is $3.73.
