Question

1) A package of 6 cellphones contains 4 that are slightly defective. Elise bought 4 of these CP's at random. Let the random variable be the number of non-defective CP's.

a) List the elements of the sample space using D and N for defective and non defective cell phones respectively.

b) Determine the values of the random variables and create a probability mass function.

c) Solve for the mean.

d) Solve for the variance.

e) Solve for the standard deviation.

Answer 1

a)

$$S=\{DDDD, DDDN, DDND, DNDD,$$

$$NDDD,DDNN,DNDN, DNND,$$

$$NDND, NDDN, NNDD\}$$

b)

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 \\ \hline p(x) & 1/11 & 4/11 & 6/11 \\ \end{array}$$

c)

$$\mu=E(X)=\dfrac{1}{11}(0)+\dfrac{4}{11}(1)+\dfrac{6}{11}(2)=\dfrac{16}{11}$$

d)

$$E(X^2)=\dfrac{1}{11}(0)^2+\dfrac{4}{11}(1)^2+\dfrac{6}{11}(2)^2=\dfrac{28}{11}$$

$$Var(X)=\sigma^2=E(X^2)-(E(X))^2=\dfrac{28}{11}-(\dfrac{16}{11})^2$$

$$=\dfrac{52}{121}$$

e)

$$\sigma=\sqrt{\sigma^2}=\sqrt{\dfrac{52}{121}}=\dfrac{2\sqrt{13}}{11}$$