代做Econ 320 Spring 2024 Problem Set 1 - Statistics Review

Econ 320 Spring 2024

Problem Set 1 - Statistics Review

The due date for this assignment is February 8 at 10:00 a.m EST.

1. [10 points] Remind yourself of the terminology we developed in Lectures 1-4. Suppose we are interested in the causal effect of military service on earnings later in life.

(a) [2 points] We run a phone survey where we ask 1,000 respondents whether they served in the military and how much they earn. The data we collected is an example of a          .

(b) [2 points] The US government has data on every person who has served in the military and tax records on earnings for every citizen of the US. This is an example of data for the          .

(c) [2 points] Suppose we take our phone survey and calculate the difference in average earnings between individuals who served in the military and those who didn’t. This would be an example of an           .

(d) [2 points] The expected difference in earnings between those who served in the military and those who didn’t would be an example of an          .

(e) [2 points] The average causal effect of serving in the military on earnings would be an example of a           .

2. [10 points] Let X be distributed as fX (x) = 2/1e −0.5x over the set of positive real numbers. Compute the E [X].

3. [15 points] Let

and

and

(a) [8 points] Compute E (X), E (Y), E (X2), E (Y2), V ar (X) and V ar (Y).

(b) [7 points] Compute E [XY ], Cov (X, Y) and Corr (X, Y).

4. [10 points] Let

with p = 1/10.

(a) [5 points] Compute E (X). Provide details of your calculation.

(b) [5 points] Compute V ar(X). Provide details of your calculation..

5. [10 points] Let X be a Bernoulli random variable with a probability of success equal to p, that is, X = {0, 1} and P (X = 1) = p, 0 < p < 1. Using the properties of expectations, compute

E [(X − E [X])4].

Please state all the steps in the calculations.

6. [10 points] Consider the two-way table of smoking status versus gender:

(a) [5 points] If one person is selected at random from the two-way table, what is P (Female), P (Smoker), P (Female and Smoker), P (Female and Non-Smoker), P (Male and Smoker), P (Male andNon-Smoker)?

(b) [5 points] Suppose the person we pick is male. What is the probability of him being a smoker? What if the selected person is female? What is the probability of her being a smoker? Are the variables smoker and gender independent?

7. [10 points] Let X and Y be independent random variables with pdf’s

and

Compute E (X), E (Y ), E [XY ], and Corr (X, Y ).

8. [10 points] The covariance between X and Y is defined as

Cov (X, Y) = E [(X − E (X)) (Y − E (Y))] .

Show that we can also write

Cov (X, Y) = E (XY) − E (X) E (Y).

9. [15 points] Let {Xi} ni=1 be a random sample from a population with mean µ and variance σ2 . Let

Show that (state all intermediate steps):

(a) [7 points] E [¯Xn] = µ.

(b) [8 points] V ar [¯Xn] = n/σ2