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Unformatted text preview: PAM 3100: Multiple Regression Analysis Mathematics and Statistics Review Fall 2010 Michael Lovenheim [email protected] The symbol Σ is the capital Greek letter sigma, and means “the sum of.” The letter i is called the index of summation . This letter is arbitrary and may also appear as t , j , or k . The expression is read “the sum of the terms x i , from i equal one to n .” The numbers 1 and n are the lower limit and upper limit of summation. 1 n i i x = ∑ n n i i x x x x x + + + + = ∑ = ... 3 1 2 1  this is the sum of the product of the variables x and y. For any constant c, and We thus can write the mean of the x i as: You can also prove that n n n i i i y x y x y x y x y x + + + + = ∑ = ... 3 3 1 2 2 1 1 ∑ ∑ ∑ = = = + = + n i n i i i i n i i y x y x 1 1 1 ) ( ∑ ∑ ∑ = = = ≠ + n i i i n i i n i i y x y x 1 1 1 ∑ ∑ ∑ = = = ≠ n i i n i i n i i i y x y x 1 1 1 ∑ ∑ = = = n i i n i i x c cx 1 1 nc c n i = ∑ = 1 ∑ = = n i i x n x 1 1 ∑ = = − n i i x x 1 ) ( Random Variables Definition: A random variable is a mapping from a set of outcomes to a set of real numbers. Example 1: flipping a coin Functions of random variables are also random variables: • Let x 1 , x 2 , x 3 …x 10 be 10 realizations of the random variable X. • Then is also a random variable. Outcome Random Variable (X) Tails Heads 1 ∑ = = 10 1 10 1 i i x X Discrete Random Variables The random variable X in the previous example is a discrete random variable. Another example is a jar filled will green, yellow and red balls. Let ½ of the balls be green, ¼ be yellow and ¼ be red. Define the random variable Y =1 if the ball is yellow, =2 if the ball is green and =3 if the ball is red. We define the Probability Distribution Function (PDF) of Y as follows: P 1 =P(Y=1)=1/4 ; p 2 =P(Y=2)=1/2 and p 3 =P(Y=3)=1/4 The PDF completely describes all values of a discrete random variable and their relative likelihoods. Continuous Random Variables A continuous random variable has an infinite number of values, so it takes on any particular value with probability zero. We describe a continuous random variable by its Cumulative Distribution Function (CDF) : CDF of random variable Z: Because CDFs always range from 0 to 1, P( Z >c)=1F(c) for any number c. Because for any number c, P(Z=c)=0, we think of probabilities of Z being between two values with continuous random variables: P(a< Z <b)=F(b)F(a), for a<b. ) ( ) ( z Z P z F ≤ = Continuous Random Variables The most famous continuous random variable is the normal distribution (more on this in a moment) Normal Probability Density Functions with Mean 0 and Variance σ 2 Features of PDFs: Expected Value The expected value ( ) of a distribution is a weighted average of all possible values of a random variable....
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This note was uploaded on 01/30/2012 for the course PAM 3100 at Cornell University (Engineering School).
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