c Larger cliques may occur involving groups of nodes of any size k Derive a

C larger cliques may occur involving groups of nodes

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c) Larger cliques may occur involving groups of nodes of any size k . Derive a general formula for the expected number of cliques of any size 2 k n as a function of the number of nodes, n . What is the expected number of cliques of size k = 4 among n = 10 people? Question 4: (40 points) We will now analyze some data collected by observing the famous “Old Faithful” geyser in Yellowstone National Park. We define random variable S to be the time an eruption lasts, and random variable T to be the “waiting time” until the next eruption. These are clearly continuous random variables, but we do not precisely know their true distribution. Instead we have a dataset with n = 272 independent observations ( s i , t i ) , i = 1 , . . . , 272, of the eruption time s i and subsequent waiting time t i . See Figure 1 for a plot of this data. In the following questions, we compute various quantities using the empirical distribution of the data. The empirical distribution of eruption time and waiting time can be represented by a probability mass function p ST ( s, t ) which places probability 1 /n on each of the n data points, and probability 0 on the continuous range of other ( s, t ) values. Under this distribu- tion, the expected values of S and T then take the following simple form: E [ S ] = 1 n n X i =1 s i , E [ T ] = 1 n n X i =1 t i . a) The variance of random variable S equals Var [ S ] = E [ S 2 ] - E [ S ] 2 . Give formulas for computing Var [ S ] and Var [ T ] under the empirical distribution. Use Python’s numpy.sum function to write your own code that computes these variances, and report their values. Hint: Various definitions of the “sample variance” can be found in statistics references, and they are not all equivalent to the variance of the empirical distribution.

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• Fall '08
• Smyth,P
• Variance, Probability theory, probability density function

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