2. Over time, I have observed my monthly spending habits follow a random variabl

Question

2. Over time, I have observed my monthly spending habits follow a random variable J which is distributed normally with mean $2000 and standard deviation $300 (or, using statistical notation, J~N(2000,90000) while my wife’s spending is a random variable M~N(500,10000).

a. If our monthly household spending is denoted as H = J + M, what is our average household spending and the variance of our household spending?

b. It turns out that I periodically buy gifts (food, housing, transportation) for my wife (which is part of my spending) and, when I do so, she has less need to make purchases. Thus, J and M are negatively correlated. Specifically, if the Cov(J,M) = -40,000, then what is the mean and variance of H?

Explanation / Answer

a) J and M are independent so their sum H is normally distributed.

With mean 2000+500=2500 and

Variance 90,000+10,000=100,000,

Thus H(2500, 100,000)

b) Since J and M are not independent in this case. So

Mean is 2000+500=2500

Variance is 90,000+10,000-40,000=60,000

Thus H(2500, 40000)

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