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Chapter 4: Q4.2-9E (page 225)

Suppose that the proportion of defective items in a large lot is p, and suppose that a random sample of n items is selected from the lot. Let X denote the number of defective items in the sample, and let Y denote the number of non-defective items. Find E (X 鈭� Y)

Short Answer

Expert verified

\(E\left( {X - Y} \right) = n\left( {2p - 1} \right)\)

Step by step solution

01

Given information

A large lot contains both defective and non-defective items.

Total number of items: n

Total number of defective items: p

X denotes the number of defective items in the sample, and let Y denote the number of non-defective items. We have to find out E (X 鈭� Y)

02

Find E (X − Y)

\(\begin{array}{c}E\left( {X - Y} \right) = E\left( X \right) - E\left( Y \right)\\ = np - n\left( {1 - p} \right)\\ = n\left( {2p - 1} \right)\end{array}\)

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Most popular questions from this chapter

Suppose that a fair coin is tossed repeatedly until a head is obtained for the first time.

(a) What is the expected number of tosses that will be required?

(b) What is the expected number of tails that will be obtained before the first head is obtained?

Consider a coin for which the probability of obtaining a head on each given toss is 0.3. Suppose that the coin is to be tossed 15 times, and let X denote the number of heads that will be obtained.

  1. What prediction of X has the smallest M.S.E?
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Suppose that the random variable X has to mean\(\mu \), and variance\({\sigma ^2}\)and that\(Y = aX + b\)Determine the values of a and b for which\(E\left( Y \right) = 0\)and\(Var\left( Y \right) = 1\)

Let X have the binomial distribution with parameters n and p. Let Y have the binomial distribution with parameters n and \(1 - p\). Prove that the skewness of Y is the negative of the skewness of X. Hint: Let \(Z = n - X\) and show that Z has the same distribution as Y.

Suppose that a point\({{\bf{X}}_{\bf{1}}}\)is chosen from the uniform distribution on the interval\(\left( {{\bf{0,1}}} \right)\)and that after the value\({{\bf{X}}_{\bf{1}}}{\bf{ = }}{{\bf{x}}_{\bf{1}}}\)is observed, a point\({{\bf{X}}_{\bf{2}}}\)is chosen from a uniform distribution on the interval\(\left( {{{\bf{x}}_{\bf{1}}}{\bf{,1}}} \right)\). Suppose further that additional variables\({{\bf{X}}_{\bf{3}}}{\bf{,}}{{\bf{X}}_{\bf{4}}}{\bf{,}}...\)are generated in the same way. Generally,\({\bf{j = 1,2,}}...{\bf{,}}\)after the value\({{\bf{X}}_{\bf{j}}}{\bf{ = }}{{\bf{x}}_{\bf{j}}}\)has been observed,\({{\bf{X}}_{{\bf{j + 1}}}}\)is chosen from a uniform distribution on the interval\(\left( {{{\bf{x}}_{\bf{j}}}{\bf{,1}}} \right)\). Find the value of\({\bf{E}}\left( {{{\bf{X}}_{\bf{n}}}} \right)\).
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