Publications by Misha Kollontai
DATA605_Final
Problem 1 Using R, generate a random variable X that has 10,000 random uniform numbers from 1 to N, where N can be any number of your choosing greater than or equal to 6. Then generate a random variable Y that has 10,000 random normal numbers with a mean of \(μ=σ=\frac{N+1}{2}\) N <- 44 n <- 10000 X <- runif(n,1,N) Y <- rnorm(n,(N+1)/2,(N+1...
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DATA605_Discussion16
APEX Calculus, p.711, Ex. 29 In Exercises 27 – 30, form a function \(z = f(x, y)\) such that \(f_{x}\) and $f_{y} match those given. \(f_{x}=6xy-4y^{2}\), \(f_{y}=3x^{2}-8xy+2\) \[\int f_{x}dx=\] \[=\int 6xy-4y^{2}dx\] \[=3x^{2}y-4xy^{2}+C\] \[\int f_{y}dy=\] \[\int 3x^{2}-8xy+2dy=\] \[3x^{2}y-4xy^{2}+2y+C\] Adding all of the unique terms we ...
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DATA_605_Discussion15
APEX Calculus, p.496, Ex. 17 Use the Taylor series given in Key Idea 8.8.1 to verify the given identity: \[cos(-x) = cox(x)\] We are given that: \[cos(x) = \sum_{n=0}^{\infty} (-1)^{n}\frac{x^{2n}}{(2n)!}\] and the first few terms are: \[1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + ...\] The first few terms of cos(-x) are then: \[...
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DATA605_Discussion14
APEX Calculus, p.194, Ex. 32 The distance, in feet, a stone drops in \(t\) seconds is given by \(d(t) = 16t^{2}\). The depth of a hole is to be approximated by dropping a rock and listening for it to hit the bottom. What is the propagated error if the time measurement is accurate to 2/10ths of a second and the measured time is: 2 seconds? 5 seco...
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DATA605_Dis13
Using R, build a multiple regression model for data that interests you. Include in this model at least one quadratic term, one dichotomous term, and one dichotomous vs. quantitative interaction term. Interpret all coefficients. Conduct residual analysis. Was the linear model appropriate? Why or why not? Upload Dataset Data taken from Kaggle Med...
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DATA605_Exercise9_Response
Response to Exercise 2 of Introduction to Probability Let \(S_{200}\) be the number of heads that turn up in 200 tosses of a fair coin. Estimate: \(E(x) = np = 200 * 0.5 = 100\) \(\sigma^{2} = \sqrt{npq}=\sqrt{200*0.5*0.5}=\sqrt{50}=5\sqrt{2}\) var <- 5*sqrt(2) \(P(S_{200} = 100)\) Equivalent to: \(1 - P(S_{200}\le99) - P(S_{200}\ge101)\) \(P(...
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DATA605_Exercise9
Exercise 11 on page 339 of Introduction to Probability Write a computer program to simulate 10,000 Bernoulli trials with probability .3 for success on each trial. Have the program compute the 95 percent confidence interval for the probability of success based on the proportion of successes. Repeat the experiment 100 times and see how many times t...
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DATA_605 Exercise8
Exercise 9 on page 290 of Introduction to Probability Prove that you cannot load two dice in such a way that the probabilities for any sum from 2 to 12 are the same. (Be sure to consider the case where one or more sides turn up with probability zero.) For each dice, there are 6 probabilities - P1, P2, P3, P4, P5 and P6. Let us assume that we can...
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DATA605_Discussion 5
Exercise 26 on page 39 of Introduction to Probability Two cards are drawn successively from a deck of 52 cards. Find the probability that the second card is higher in rank than the first card. Hint: Show that 1 = \(P(higher) + P(lower) + P(same)\) and use the fact that \(P(higher) = P(lower)\) Clearly, there are 3 possibilities when drawing two c...
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DATA606_HW9
Baby weights, Part I. (9.1, p. 350) The Child Health and Development Studies investigate a range of topics. One study considered all pregnancies between 1960 and 1967 among women in the Kaiser Foundation Health Plan in the San Francisco East Bay area. Here, we study the relationship between smoking and weight of the baby. The variable smoke is c...
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