MAPOA INSTITUTE OF TEGHNOLOGY Dep<rrtment of Mothemqtio vtst0N The l‚4apua lnstitute of Technology shall be a global center of excellence in education by providing instructions that are current in content and state-of-the-art in delivery; by engaging in cutting-edge‚ high impact research; and by aggressively taking on presen!day global concerns. Mrssr0N a. b. c. d. The [/apua Institute of Technology disseminates‚ generates‚ preserves and applies knowledge in various fields of study. The lnstitute
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version: September 14‚ 2009 1 Historical introduction The history of copulas may be said to begin with Fr´ chet [69]. He studied the fole lowing problem‚ which is stated here in dimension 2: given the distribution functions F1 and F2 of two random variables X1 and X2 defined on the same probability space Ω ‚F‚P ‚ what can be said about the set Γ F1 ‚F2 of the bivariate d.f.’s whose marginals are F1 and F2 ? It is immediate to note that the set Γ F1 ‚F2 ‚ now called the Fr´ chet class of F1 and F2
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Probability 2 Theory Probability theory is the branch of mathematics concerned with probability‚ the analysis of random phenomena. (Feller‚ 1966) One object of probability theory is random variables. An individual coin toss would be considered to be a random variable. I predict if the coin is tossed repeatedly many times the sequence of it landing on either heads or tails will be about even. Experiment The Experiment we conducted was for ten students to flip a coin one hundred times
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of the binomial distribution (q + p)2 is 2p. 12. Show that mean is 2p and σ2 = 2pq for a binomial distribution in which n = 2. 13. A random variable X has a binomial distribution with E(X) = 2.4 and p = 0.3. Find the standard deviation of X. 14. Describe the normal distribution and write down its equation. 15. What is standard normal variable? 16. The value 2nd moment about mean in a normal distribution is 5. Find the 3rd and 4th moment about mean for this distribution. 17
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Mathematical Systems Probability Solutions by Bracket A First Course in Probability Chapter 4—Problems 4. Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are alike and all 10! possible rankings are equally likely. Let X denote the highest ranking achieved by a woman (for instance‚ X = 1 if the top-ranked person is female). Find P X = i ‚ i = 1‚ 2‚ 3‚ . . . ‚ 8‚ 9‚ 10. Let Ei be the event that the the ith scorer is female. Then the
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University of Phoenix Material Learning Team Summary Worksheet TWO (due week Five) Brenda Rivera As a learning team‚ complete the table with formulas‚ rules‚ and examples from each section of Chapters 4‚ 5‚ 6‚7‚8‚9‚10 and 11 in the textbook. The completed summary will help prepare you for the Final Exam in Week 5. Points will be awarded for completion of the project. Study Table for Weeks One and Two Chapter 4 Systems of Linear Equations; Matrices (Section 4-1 to 4-6)
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STA304 H1 S/1003 H S Winter 2013 Dragan Banjevic (I) Note: A lot of material will be used from Internet‚ some with reference‚ some without. 2 CITY OF TORONTO NEIGHBOURHOODS 1 West Humber-Clairville 19 Long Branch 36 Newtonbrook West 54 O’Connor-Parkview 2 Mount Olive-SilverstoneJamestown 20 Alderwood 37 Willowdale West 55 Thorncliffe Park 3 Thistletown-Beaumond Heights 21 Humber Summit 38 Lansing-Westgate 56 Leaside-Bennington 4 Rexdale-Kipling
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P(X > 8) This will become P(Y < 10) AND Y~B(10‚0.1) Poisson Distribution Binomial probability distribution is defined as: * P(X=r) = e-λ x λrr! * Distribution is written as: X~Po(λ) Conditions include: * Events occur at random * All events are independent of one another * Average rate of occurrence remains constant * Zero probability of simultaneous occurrences E(X) = λ Var(X) = λ SD(X) = Var (X) = λ To calculate the probabilities: * P(X
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The population standard deviation σ of a discrete random variable ‚ Measure how close a random variable tends to be the population mean μ‚ so you must understand μ before you understand σ If you have a random variable like a bet at a casino or and investment then the standard deviation σ measure the risk‚ if there is a lot of risk then the standard deviation is high The formulas for standard deviation are given below but you should look at the examples first Population mean Population variance
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Name: ID: Homework 3 Solutions 1. [§6-4] Let X1 ‚ X2 ‚ . . . ‚ X8 be i.i.d. normal random variables with mean µ and standard deviation σ. Define X −µ ‚ T =√ S 2 /n where X is the sample mean and S 2 is the sample variance. (a) Find τ1 such that P(|T | < τ1 ) = .9; and (b) find τ2 such that P(T > τ2 ) = .05. (a) First‚ we notice that T follows a t7 distribution. Since the t distribution is symmetric about 0‚ we have 0.9 = P(|T | < τ1 ) = P(T < τ1 ) − P(T < −τ1 ) = 2P(T < τ1 ) − 1. Thus P(T < τ1
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