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Stat 401: Applied Statistical Methods II Fall 2016 Homework**More:**

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**1. An exam has two questions. Let (X, Y) be the outcomes for a randomly selected student.Stat 401: Applied Statistical Methods II**

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**Fall 2016**

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**Homework #3**

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**Due Date: October 7 (in your lab)**

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**1. An exam has two questions. Let (X, Y) be the outcomes for a randomly selected student.**

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**X = 1 if the student answers question 1 correctly; and X = 0 if the student does not. Similarly, Y**

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**denotes the corresponding outcome for question 2. The joint pmf of (X, Y) is given below.**

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**From the marginal distribution, you can get the following: E(X) = 0.5 and Var(X) = 0.25.**

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**y x 0**

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**1**

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**Marginal**

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**distribution of Y 0 1 0.4**

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**0.1 0.1**

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**0.4 Marginal**

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**distribution of X**

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**0.5**

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**0.5 0.5 0.5 1.0 a) What is correlation between X and Y?**

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**b) The exam scores are determined as follows: the student gets 50 points for answering a**

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**question correctly and zero otherwise. There are no partial credits. Let S be the total score**

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**that the student will get in the exam. What are the mean and variance of S? (Do not**

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**compute this from the distribution of S. Use your answers in parts (a) and (b) and the other**

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**information you are given to answer this question.)**

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**c) Write down the pmf of S.**

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**2. Joe Smith plays a betting game with his friends. He puts in one dollar to join the game. If he wins**

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**the game, he gets $2. So his gain is $1. If he loses, he gets nothing, so his gain is ?$1. Let V be**

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**the random variable that denotes the gain (amount of money in dollars) made by Joe Smith.**

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**(Remember that a loss is counted as negative gain). The pmf of V is given below.**

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**v**

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**p(v) a)**

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**b)**

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**c)**

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**d) -1**

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**0.6 1**

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**0.4 Consider the random variable V2 given by V times V. Write down the joint pmf of (V, V2).**

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**Are V and V2 independent? Why or why not?**

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**Consider the random variable V3 given by V times V times V. What is the joint pmf of (V, V3)?**

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**What is the correlation between V and V3? [Hint: You can answer this question without**

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**computing the covariance of V and V3 but you must explain clearly.] 1 3. Consider again the game in Question 2. Joe Smith is planning to play the game 100 times. The**

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**100 outcomes will be independent of each other.**

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**a) What is his expected total gain if he plays 100 times?**

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**b) What is the variance of his total gain?**

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**c) Use the Central Limit Theorem to compute the probability that his total gain will be positive.**

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**4. The distribution of piston diameters is normally distributed with mean ? inches and standard**

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**deviation 5 inches. Engineer Dan takes a sample of 25 pistons from the manufacturing assembly**

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**line. Let X1, X2, ?, X25 be denote these observations which are iid.**

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**a. What is the probability that Xbar, the sample mean of Dan?s 25 observations, is within 1**

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**inch of the population mean ?.**

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**b. Suppose engineer Jane decides to take a sample of 30 observations. Let Y1, Y2, ?, Y30**

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**denote these observations which are also iid. Jane computes the probability that her**

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**sample mean Ybar is within 1 inch of the population mean ?. Without doing any**

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**calculations, do you think this probability will be smaller or bigger than that for Dan?s**

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**sample mean? Explain your answer clearly.**

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**5. Consider two random variables X and Y with variances 4 and 9 respectively.**

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**a. What is the variance of (X ? Y) when cor (X, Y) = 0?**

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**b. What is the variance of (X ? Y) when cor(X, Y) = 0.5?**

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**c. What is the variance of (X ? Y) when cor(X, Y) = ? 0.5?**

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**6. An appliance dealer sells three different models of freezers with advertised capacity of 13, 16**

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**and 20 cubic feet. Let X = the amount of storage space of the freezer purchased by the next**

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**customer. Suppose X has the pmf given below:**

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**x**

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**p(x) 13**

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**0.2 16**

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**0.5 20**

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**0.3 a. The price of a freezer having cubic capacity X is given by 25X ? 20. What are the**

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**expected value and variance of the price paid by the next customer?**

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**b. Suppose that although the advertised capacity of a freezer is X, the actual capacity is**

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**h(X) = X ? 0.01 X2.**

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**What are the expected value and variance of h(X), the actual capacity of the freezer**

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**bought by the next customer? 2**

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This question was answered on: * Feb 21, 2020 *

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