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

C larger cliques may occur involving groups of nodes

This preview shows page 2 - 3 out of 3 pages.

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.
Image of page 2

Subscribe to view the full document.

Image of page 3
  • Fall '08
  • Smyth,P
  • Variance, Probability theory, probability density function

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes