Publications by Vinayak Kamath
Data605-Week6-HomeWork6-kamath
Home Work 06 box contains 54 red marbles, 9 white marbles, and 75 blue marbles. If a marble is randomly selected from the box, what is the probability that it is red or blue? Express your answer as a fraction or a decimal number rounded to four decimal places. red_marbles <- 54 white_marbles <- 9 blue_marbles <- 75 total_marbles <- red_marbl...
4700 sym R (4846 sym/33 pcs)
Data605-Week7-HomeWork7-kamath
Home Work 07 Let X1, X2, . . . , Xn be n mutually independent random variables, each of which is uniformly distributed on the integers from 1 to k. Let Y denote the minimum of the Xi’s. Find the distribution of Y . ==> The solution should be: \(1≤j≤k,m(j)= ( ((k−j+1)^n−(k−j)^n) / k^n )\) Since Y is the minimum value of Xi over all of...
2863 sym R (1666 sym/24 pcs)
Data605-Week8-HomeWork8-kamath
Home Work 08 A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the first of these bulbs to burn out? ==> Solution: E[Xi] = 1 / λi = 1000 Expected lifetime of a bulb is 1000 hours. λi = 1 / 1000 Xi is exponential so min{X1,X2,…,X100} ∼ exponential (∑ i=1 to 100 for λi) E...
845 sym R (425 sym/8 pcs) 1 img
Data605-Week9-HomeWork9-kamath
Home Work 09 The price of one share of stock in the Pilsdorff Beer Company (see Exercise 8.2.12) is given by Yn on the nth day of the year. Finn observes that the differences Xn = Yn+1 − Yn appear to be independent random variables with a common distribution having mean μ = 0 and variance sigma^2 = 1/4. If Y1 = 100, estimate the probability t...
1804 sym R (722 sym/6 pcs)
Data605-Week9-Discussion9-kamath
CHAPTER 9. CENTRAL LIMIT THEOREM Let X be a continuous random variable with mean μ(X) and variance σ^2(X), and let X^∗=(X−μ)σ be its standardized version. Verify directly that μ(X^∗)=0 and σ2(X∗)=1. \[X^∗= \frac{(X−μ)}{σ} \] \[E(X^∗)=\frac{1}{σ}(E(X)−μ) \] \[E(X^∗)=\frac{1}{σ}(μ−μ) \] \[E(X^∗)=μ(X^∗)=0 \] \[...
394 sym
Data605-Week16-Discussion16-Kamath
Functions of Several Variables Exercise 12.1 #17 In Exercises 15 – 22, describe in words and sketch the level curves for the function and given c values. \[ f(x,y) = x - y^2; c = -2; 0; 2 \] ==> Consider \(c=0\). Then \(f(x,y)= x - y^2 = c = 0\), so \(y= ±\sqrt{x}\). With \(c=−2\), \(y=±\sqrt{x+2}\), and with \(c=2\), \(y=± \sqrt{x-2}\)...
417 sym R (339 sym/3 pcs) 1 img
Data605-Week15-Discussion15-Kamath
Sequences and Series Exercise 8.8 #5 Key Idea 8.8.1 gives the nth term of the Taylor series of common functions. In Exercises 3 – 6, verify the formula given in the Key Idea by finding the first few terms of the Taylor series of the given function and identifying pattern. \[ f(x) = \frac{1}{(1 - x)}; c = 0 \] ==> \[ f′(x)=\frac{1}{(1 - x)^2...
546 sym R (310 sym/5 pcs)
Data605-Week14-HomeWork14-kamath
Home Work 14 - TAYLOR SERIES Q. This week, we’ll work out some Taylor Series expansions of popular functions. \(f(x) = \frac{1}{1-x}\) \(f(x) = e^x\) \(f(x) = ln(1 + x)\) For each function, only consider its valid ranges as indicated in the notes when you are computing the Taylor Series expansion. Please submit your assignment as a R-Markdo...
1984 sym R (390 sym/6 pcs)
Data605-Week13-Discussion13-kamath
Discussion 13 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? For this excercise have pick...
1383 sym R (4409 sym/14 pcs) 6 img
Data605-Week11/12-Discussion11/12-kamath
Discussion 11 and 12 Using R, create a simple linear regression model and test its assumptions. You may use any data that interest you. For this excercise have picked the internal dataset - trees #Displaying the trees dataset contents: str(trees) ## 'data.frame': 31 obs. of 3 variables: ## $ Girth : num 8.3 8.6 8.8 10.5 10.7 10.8 11 11 11...
1196 sym R (2100 sym/13 pcs) 5 img