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Probability and Statistical Inference | 9th Edition | ISBN: 9780321923271 | Authors: Robert V. Hogg, Elliot Tanis, Dale Zimmerman
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Let W equal the weight of laundry soap in a 1-kilogram box

Chapter 4, Problem 3E

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QUESTION:

PROBLEM 3E

Let W equal the weight of laundry soap in a 1-kilogram box that is distributed in Southeast Asia. Suppose that P(W < 1) = 0.02 and P(W > 1.072) = 0.08. Call a box of soap light, good, or heavy depending on whether {W < 1}, {1 ≤ W ≤ 1.072}, or {W > 1.072}, respectively. In n = 50 independent observations of these boxes, let X equal the number of light boxes and Y the number of good boxes.

(a) What is the joint pmf of X and Y?

(b) Give the name of the distribution of Y along with the values of the parameters of this distribution.

(c) Given that X = 3, how is Y distributed conditionally?

(d) Determine E(Y |X = 3).

(e) Find ρ, the correlation coefficient of X and Y.

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QUESTION:

PROBLEM 3E

Let W equal the weight of laundry soap in a 1-kilogram box that is distributed in Southeast Asia. Suppose that P(W < 1) = 0.02 and P(W > 1.072) = 0.08. Call a box of soap light, good, or heavy depending on whether {W < 1}, {1 ≤ W ≤ 1.072}, or {W > 1.072}, respectively. In n = 50 independent observations of these boxes, let X equal the number of light boxes and Y the number of good boxes.

(a) What is the joint pmf of X and Y?

(b) Give the name of the distribution of Y along with the values of the parameters of this distribution.

(c) Given that X = 3, how is Y distributed conditionally?

(d) Determine E(Y |X = 3).

(e) Find ρ, the correlation coefficient of X and Y.

ANSWER:

Step 1 of 6:

Given that let W equal the weight of laundry soap in a 1-kilogram box .

Also it is given that P(W<1)=0.02 and P(W>1.072)=0.08. A box of soap is called light if (W<1), soap box is called good if (1W1.072) and heavy if (W>1.072).

Let X denotes the number of light boxes and Y denotes the number of good boxes. The total number of independent observations n=50.

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