Econ 103经济代写

ECON 103：宏观经济原则

Introduction to the theory of determination of total or aggregate income, employment, output, price levels, and the role of money in the economy. Primary emphasis on monetary and fiscal policy, inflation, unemployment, economic growth, and international economics. 3 undergraduate credit hours

1. Suppose I know the joint distribution of $X$ and $Y$. Using the joint pdf of $X$ and $Y$ I calculate the covariance between $Y$ and $X$ using the following formulas:
\begin{aligned} \operatorname{Cov}(X, Y) &=\mathbb{E}[Y X]-\mathbb{E}[Y] \mathbb{E}[X]=30 \\ \operatorname{Var}(X) &=\mathbb{E}\left[X^{2}\right]-(\mathbb{E}[X])^{2}=100 \\ \operatorname{Var}(Y) &=\mathbb{E}\left[(Y-\mathbb{E}[Y])^{2}\right]=4 \end{aligned}
Have I made any errors?
(a) No, all formulas are correct and there are no apparent errors in calculation.
(b) Yes, the formula for $\operatorname{Var}(Y)$ is incorrect.
(c) Yes, the formula for $\operatorname{Cov}(X, Y)$ is incorrect.
(d) Yes, the formula for $\operatorname{Var}(X)$ is incorrect.
(e) Yes, all formulas are correct but I have made an error when computing either the variances or the covariance.

Proof .

Answer: All formulas are correct. However, recall from Homework that, by the Cauchy-Schwarz inequality the correlation coeffecient must be bounded between $-1$ and $1 .$ Here the correlation coeffecient is given by
$$\frac{\operatorname{Cov}(X, Y)}{\sqrt{\operatorname{Var}(X)} \sqrt{\operatorname{Var}(Y)}}=\frac{30}{10 \cdot 2}=1.5 .$$
So, I must have made a calculation error.

Suppose that I can pay $\$ 0.8$to enter a lottery whose payout (in dollars) is described by a random variable$Y$with pdf: $$f_{Y}(a)= \begin{cases}c a^{2} & \text { if } a \in[0,1] \\ 0 & \text { otherwise }\end{cases}$$ for some unknown constant$c>0$. Should I pay to enter this lottery? (a) It is impossible to tell because we do not know$c$. (b) Yes, because the expected value,$\mathbb{E}[Y]$, is positive. (c) Yes, because the expected value,$\mathbb{E}[Y]$, is greater than$\$0.8$.
(d) No, because the expected value, $\mathbb{E}[Y]$, is negative.
(e) No, because the expected value, $\mathbb{E}[Y]$, is less than $\$ 0.8$. Proof . Answer: Using the fact that the pdf integrates to one, we can solve for$c$: $$\int_{0}^{1} c a^{2} d a=\left.1 \Longrightarrow c \cdot \frac{a^{3}}{3}\right|_{0} ^{1}=1 \Longrightarrow \frac{c}{3}=1 \Longrightarrow c=3$$ Now, we can solve for the exptected value of$Y\$ :
$$\mathbb{E}[Y]=\int_{0}^{1} 3 f_{Y}(a) \cdot a d a=\int_{0}^{1} 3 a^{3} d a=\left.3 \cdot \frac{a^{4}}{4}\right|_{0} ^{1}=\frac{3}{4}$$
Because the expected value of this lottery is lower than the cost to enter it, we should not enter the lottery.

## Math 152 lab

math2030代写

Faculty Teaching the Course:

This course is taught by the below faculty~ you may click on their name to view their website with additional information.  Please check the Course Explorer or Enterprise/Self-Service to see what section they will be teaching (teaching schedules vary by semester).
Eric McDermott
Joe Petry

### Past Course Syllabi:

The following syllabi are from past semesters and should only be used as a guide for the information covered in the course and general structure of the course. The instructors have the right to change the course for upcoming semesters ~ please refer to the syllabus they distribute the first day of class.
Econ 103 Macroeconomic Principles Eric McDermott Past Syllabus

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https://courses.illinois.edu/schedule/terms/ECON/103

### Section Information:

• AL1 or AL2 Lecture MUST be paired with an AQ Discussion (Quiz) Section, and they must be added at the same time when registering. This is in-class for both the lecture twice per week and the discussion.
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