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Homework1-key - University of California Davis ARE 106...

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University of California, Davis ARE 106 - Winter 2011 Problem Set 1 The quiz based on this homework will be held at the beginning of class, Thursday, January 13, 2011. Conceptual Problems 1. What is a random variable? A random variable is a variable whose value is unknown until it is observed. A variable that is not perfectly predictable. 2. What do we mean when we talk about the expected value of the random variable, X? (Hint: look at page 487.) The expected value of X is a weighted average of its values. If we were to have an infinite number of realizations of a random variable, E(X) would be the average of that infinite series of values. 3. In words, what is the difference between the expected value of a random variable and its sample mean? The difference can be seen in the formula for the expected value compared to the formula for the sample mean. An expected value weights each possible value of the random variable by the probability that it will occur. The sample mean weights each observed value of the random variable by the proportion of times in the sample that is is observed. 4. Give an example of: (a) A discrete random variable. Anything that can take on a finite number of values. For example, how many days in a week do I ride my bike to campus x=0,1,2,3,4,5. (b) A binary random variable. Anything with only two possible outcomes, coded 0 or 1. A coin toss is a common example x=0 if heads, x=1 if tails. Another example is “Did the sun come up this morning?” x=0 if no, x=1 if yes. (c) A continuous random variable. Any variable that can take on infinite number of values. Height is one example. You can be 5’4” or 5’5” or any one of an infinite number of possibilities in between (e.g. 5’4.584082347238) Computational Questions 1. State the following in summation notation: (a) x 1 + x 2 + x 3 + x 4 + x 5 5 X i =1 x i 1
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University of California, Davis ARE 106 - Winter 2011 (b) x 1 y 1 + x 2 y 2 + x 3 y 3 + x 4 y 4 4 X i =1 x i y i (c) x 13 + x 14 + x 15 15 X i =13 x i (d) f ( x 1 ) + f ( x 2 ) + f ( x 3 ) + f ( x 4 ) + . . . + f ( x n ) n X i =1 f ( x i ) (e) x 1 y 1 + x 1 y 2 + x 1 y 3 + x 1 y 4 + x 2 y 1 + x 2 y 2 + x 2 y 3 + x 2 y 4 4 X j =1 2 X i =1 x i y j 2. Write out the following sums: (a) 3 i =1 x i - a x 1 - a + x 2 - a + x 3 - a = x 1 + x 2 + x 3 - 3 a (b) 6 i =1 a a + a + a + a + a + a (c) 3 i =1 ( x i - a ) 2 ( x 1 - a ) 2 + ( x 2 - a ) 2 + ( x 3 - a ) 2 (d) 3 j =1 2 i =1 x i y j x 1 y 1 + x 1 y 2 + x 1 y 3 + x 2 y 1 + x 2 y 2 + x 2 y 3 3. Explain why the following is true: ( n X i =1 x i ) 2 6 = n X i =1 x 2 i The left-hand-side term is ( x 1 + x 2 + x 3 + . . . + x n )( x 1 + x 2 + x 3 + . . . + x n ) while the right-hand-side term is x 2 1 + x 2 2 + x 2 3 + . . . + x 2 n . The left term, when you multiple it out, has all the square terms but also a whole bunch of cross terms: x 2 1 + x 2 2 + x 2 3 + . . . + x 2 n + 2 x 1 x 2 + 2 x 1 x 3 . . . + 2 x 1 x n + 2 x 2 x 3 . . .
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