Financial statements analysis for Tesla Inc. for for 4 since 2016 to today(cash flow statement, balance sheet and income statement) PDF

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Financial statements analysis for Tesla Inc. for for 4 since 2016 to today(cash flow statement, balance sheet and income statement)...

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STAT 3701 Homework 1 Fall 2021 September 7, 2021 Show all work. Submit your solutions in a pdf document on Canvas. Include your R code (which must be commented and properly indented) in the pdf file. We also recommend you submit one text file (.txt) with all your R code (comments and all) clearly labeled with the problem it goes with. This must be properly indented. Before every solution with random sampling use set.seed(3701).

Question 1 (8 points.) Let X ∼ Binom(θ ). Derive E[X 3 − 3X 2 + 2X ] assuming n ≥ 3.

Question 2 You are going to the lake to try and catch a big fish. Let X be the random variable representing the weight of the fish you catch and let X ∼ N(30, 5). The lake mandated that you can only keep a fish if it weighs more than 32 pounds and you decide you will not leave until you can bring a fish home. Assume the fish you catch are independent of each other. (a) (4 points) Use pnorm , what is the probability for each pull that you can catch a fish you can take home? (b) (4 points) Let Y be the number of pulls you need to have till you catch a fish and can bring it home. Derive the probability mass function (pmf) of Y . (c) (6 points) Now before you depart, you are interested in how many pulls on average you are expected to have so that you can bring a fish home (i.e., the expected value of Y) and you want to calculate this expected number by simulation. 1. You can use while loop to simulate a day at the lake. This loop should output the number of pulls it take to take a fish home, which is one realization of Y . 2. Repeat 1 for reps times (here let reps = 500), so that we get reps iid realizations of Y . Use those iid realizations of Y to approximate E(Y ).

Question 3 A more general form of Cauchy distribution is defined by the density function f (x; m, γ) =



πγ 1 +

1

1 

x−m γ

2  ,

where m is the location parameter, γ is the scale parameter and they are both constants. We will create a function to simulate draws of a Cauchy(m, γ) distribution in this exercise using the inversion method. (a) (4 points) Derive the cdf, and the inverse function of cdf for Cauchy(m, γ). Describe a procedure to generate independent observations from a Cauchy(m, γ) distribution. (b) (5 points) Write R code that defines a function calledmyrcauchy to implement inverse method you described in 1. Only calls to R’s standard uniform generator runif are permitted. The function should have three parameters: • n, the sample size. • m, the user-specified location parameter. • γ, the user-specified scale parameter. (c) (5 points) Test your myrcauchy function by generating a generalization of random sample of size 1000 from the Cauchy(2, 1) distribution. Create a QQ-plot to compare the data percentiles to the theoretical percentiles. Only use the 1st, 2nd, 3rd, . . . , 99th percentiles (so the plot has 99 points). Please comment on the plot.

Question 4 Suppose we are interested in calculating the following integral using the Monte Carlo method: Z

2 /2

sin(x)e−x

dx.

(a) (4 points) Briefly describe a procedure to approximately calculate the above integral using draws from the standard Normal distribution. (b) (5 points) Write an R function named mc.integral to implement your method described in 1 to calculate the integral. Only calls to R’s standard uniform generator runif are permitted. Your function should contain 1 argument: n, the number of draws used to approximate the integral. The output of the function is the approximated value of the integral. (c) (5 points) Test your function mc.integral by calculating the integral with n taking values in {100, 110, 120, . . . ,4990, 5000}. Make a plot ofn against the corresponding approximated value of the integral. Comment on the plot.

Bonus question Let X be a random variable with certain distribution. The moment-generating function (mgf) of X is defined as MX (t) := E(etX ), t ∈ R if this expectation exists. (a) (5 points) Suppose X ∼ N(µ, σ 2 ), derive the moment-generating function of X .

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(b) (5 points) A very nice property of moment-generating function is that it uniquely determines the distribution, i.e., if X ’s moment-generating function exists, then it is unique. Now, now we will use this property to show that a linear combination of independent normal random variables is still normal. More specifically, let Y1 ∼ N(µ1 , σ 21 ), Y2 ∼ N(µ2 , σ 22 ) , Y1 and Y2 are independent. Let W = aY1 + bY2 , where a, b are constants. Show that W ∼ N(aµ1 + bµ2 , a2 σ12 + b2 σ22 ). (Hint: you may calculate the mgf of W and match that with the mgf of N(aµ1 + bµ2 , a2 σ12 + b2 σ22).)

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