We are interested to understand the relationship between the stock price of a listed

Subject: Mathematics / Statistics
Question
Problem 1: We are interested to understand the relationship between the stock price of a listed
company and the S&P500 Index. We collect prices for the two assets from Yahoo!Finance.

We first create the returns of the stock and the index, where the return is defined here as the daily
percentage change. We then create the following two variables:

Y: equal to 1 if the stock return in day t is larger or equal to zero, and equal to 0 if the
stock return was negative
X: equal to 1 if the market return in day t is larger or equal to zero, and equal to 0 if the
market return was negative

The joint probability density function of X and Y is given in the following Table:
Y=0 Y=1
X = 0 0.307 0.167
X = 1 0.179 0.347

Answer the following questions:

a. What does each number in the four cells represent?
b. Calculate the marginal distribution of X and Y
c. Calculate the probability density function of Y conditional on X=0 and X=1
d. Calculate the expected value of Y conditional on X=0
e. Calculate the expected value of Y conditional on X=1
f. Are X and Y independent? Justify your statement

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