A machine produces 0, 1 or 2 defective pieces in a day with an as

| A machine produces 0, 1 or 2 defective pieces in a day with an associated probability of \(\frac{{1}}{{6}}\), \(\frac{{2}}{{3}}\) and \(\frac{{1}}{{6}}\) respectively. The mean value and the variance of the number of defective pieces produced by the machine in a day, respectively, are

A. 1 and \(\frac{{1}}{{3}}\)

B. \(\frac{{1}}{{3}}\) and 1

C. 1 and \(\frac{{4}}{{3}}\)

D. \(\frac{{1}}{{3}}\) and  \(\frac{{4}}{{3}}\)

Please scroll down to see the correct answer and solution guide.

Right Answer is: A

SOLUTION

Concept:

Mean (μ) = E(x) = ∑ x P(x)

Calculation:

Let ‘x’ be the number of defective pieces

x	0	1	2
P(x)	\(\frac{{1}}{{6}}\)	\(\frac{{2}}{{3}}\)	\(\frac{{1}}{{6}}\)

Mean (μ) = E(x) = ∑ x P(x)

\(= 0 \times \frac{1}{6} + 1 \times \frac{2}{3} + 2 \times \frac{1}{6}\)

= 1

E(x²) = ∑ x² P(x)

\(= 0 \times \frac{1}{6} + 1 \times \frac{2}{3} + 4 \times \frac{1}{6}\)

= 4/3

Variance v(x) = E(x²) – [(E(x)]²

\(= \frac{4}{3} - 1 = \frac{1}{3}\)