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Unformatted text preview: Math 16b Thomas Scanlon Autumn 2010 Thomas Scanlon () Math 16b Autumn 2010 1 / 44 The mean The mean of a sequence of numbers a 1 , a 2 ,..., a n is the average: μ = 1 n ( a 1 + ... + a n ) = 1 n n X i = 1 a i Thomas Scanlon () Math 16b Autumn 2010 2 / 44 Example Let us compute the mean of a sequence of twenty rolls of a die. Thomas Scanlon () Math 16b Autumn 2010 3 / 44 Variance The variance measures the extent to which individual data points di er from the mean. As with the sum of squares of errors we used to measure the t of a recursion line to a data set, the variance is de ned as the average of the sum of squares of errors where we treat the di erence between a data point and the mean as an error . Formally, the variance, σ 2 , of the sequence of numbers a 1 ,..., a n having mean μ is σ 2 := 1 n n X i = 1 ( a i μ ) 2 The number σ := √ σ 2 is called the standard deviation . Thomas Scanlon () Math 16b Autumn 2010 4 / 44 Example With the data from our rolls of the dice, compute the variance. Thomas Scanlon () Math 16b Autumn 2010 5 / 44 Computing mean and variance How would we compute the mean and the variance of these data? There are too many data points for it to be reasonable to sum them one by one. Thomas Scanlon () Math 16b Autumn 2010 6 / 44 Frequency table We may present our data as a frequency table rather than as a list. Given a list of numbers a 1 ,..., a n taking possible values v 1 ,..., v m we de ne the relative frequency of the value v to be the number of data points a i for which a i = v divided by n . Conventionally, this is written as p j := the number of indices i for which a i = v n Note ∑ m j = 1 p j = 1 and 0 ≤ p j ≤ 1 for every j . Thomas Scanlon () Math 16b Autumn 2010 7 / 44 Examples Compute the frequency tables for our two data sets. Thomas Scanlon () Math 16b Autumn 2010 8 / 44 Random Variables We may organize the information from a relative frequency table into a function, called a random variable . Given a set of possible values V and a sequence of numbers a 1 ,..., a n from V , the random variable X corresponding to this sequence is the function de ned by X ( v ) := the relative frequency of the value v . More generally, a random variable X (on V ) is a function with domain V having the properties: ≤ X ( v ) ≤ 1 for all v in V ∑ v in V X ( v ) = 1 Thomas Scanlon () Math 16b Autumn 2010 9 / 44 Example Express our frequency table as a random variable. Thomas Scanlon () Math 16b Autumn 2010 10 / 44 Expected value One may compute the mean of a data set from its corresponding random variable. (Called in this case the expected value of X , or E ( X ) ). Let a 1 ,..., a n be a sequence of numbers with corresponding random variable X and possible values v 1 ,..., v m ....
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 Fall '06
 Sarason
 Math, Calculus, Normal Distribution, Probability theory, probability density function, Thomas Scanlon

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