Question

Company f sells fabrics known as fat quarters, which are rectangles of fabric created by cutting a yard of fabric into four pieces. Occasionally the manufacturing process results in a fabric defect. Let the random variable x represent the number of defects on a fat quarter created by company f. The following table shows the probability distribution of x.

Answer 1

By using the concept of expectation, it can be calculated that

a) Probability distribution of Y

P(Y = 0)  = 0.63

P(Y = 1) = 0.25

P(Y = 2) = 0.12

b) Mean of Y = 0.49

  Standard deviation of Y = 0.6999

c) Mean of selling price = $4.4

   Standard deviation of selling price = $0.99

What is expectation?

At first it is important to know about probability of an event.

Probability gives us the information about how likely an event is going to occur

Probability is calculated by Number of favourable outcomes divided by the total number of outcomes.

Probability of any event is greater than or equal to zero and less than or equal to 1.

Probability of sure event is 1 and probability of unsure event is 0.

Suppose x is a random variable with the probability function f(x). suppose

$x_1, x_2, ...., x_n$ are the values corrosponding to the actual occurance of the event and $p_1, p_2,...., p_n$ be the corrosponding probabilities.

Expectation is given by the formula

$E(x) = p_1x_1 + p_2x_2 + ... + p_nx_n$

a)If a fat quarter has more than 2 defects, it cannot be sold and is discarded.

The table for the probability distribution of X is given

P(X $\leq$ 2) = 0.58 + 0.23 + 0.11 = 0.92

For probability distribution of Y

P(Y = 0) = P(X= 0 | X $\leq$ 2) = $\frac{0.58}{0.92}$ = 0.63

P(Y = 1) = P(X= 1 | X $\leq$ 2) = $\frac{0.23}{0.92}$ = 0.25

P(Y = 2) = P(X= 2 | X $\leq$ 2) = $\frac{0.11}{0.92}$ = 0.12

Mean of Y = 0 $\times$ 0.63 + 1  $\times$ 0,25 + 2  $\times$ 0.12 = 0.49

E($Y^2$) = 0 $\times$ 0.63 + 1  $\times$ 0.25 + 4  $\times$ 0.12 = 0.73

Variance = $0.73 - 0.49^2$ = 0.4899

Standard deviation = $\sqrt{0.4899} = 0.6999$

b) Here, E(G) = 0.40, V(G) = $0.66^2$

The fat quarters sell for $5.00 each but are discounted by $1.50 for each defect found.

Selling price = 5 -1.5G

Mean of selling price = 5 - 1.5$\times$ 0.4 = 5 - 0.6 = $4.4

Variance of selling price = $5 - 1.5^2 \times 0.66^2$ = $0.9801

Standard deviation of selling price = $\sqrt{0.9801}$ = $0.99

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Complete Question

Company f sells fabrics known as fat quarters, which are rectangles of fabric created by cutting a yard of fabric into four pieces. Occasionally the manufacturing process results in a fabric defect. Let the random variable x represent the number of defects on a fat quarter created by company f. The following table shows the probability distribution of x.

x                   0            1            2     3         4 or more

probability 0.58 0.23 0.11 0.05 0.03

If a fat quarter has more than 2 defects, it cannot be sold and is discarded. Let the random variable Y represent the number of defects on a fat quarter that can be sold by Company F.

(a) Construct the probability distribution of the random variable Y.

(b) Determine the mean and standard deviation of Y. Show your work.

Company G also sells fat quarters. The mean and standard deviation of the number of defects on a fat quarter that can be sold by Company G are 0.40 and 0.66, respectively. The fat quarters sell for $5.00 each but are discounted by $1.50 for each defect found.

(c) What are the mean and standard deviation of the selling price for the fat quarters sold by Company