You represent a loss as a ______ amount positive negative
2 answers:
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a. I assume the following definitions for covariance and correlation:
Recall that
where is the joint density, which allows us to easily compute the necessary expectations (a.k.a. first moments):
Also, recall that the variance of a random variable is defined by
We use the previous fact to find the second moments:
Then the variances are
Putting everything together, we find the covariance to be
and the correlation to be
b. To find the conditional expectations, first find the conditional densities. Recall that
where is the conditional density of
given
, and
is the marginal density of
.
The law of total probability gives us a way to obtain the marginal densities:
Then it follows that the conditional densities are
Then the conditional expectations are
c. I don't know which theorems are mentioned here, but it's probably safe to assume they are the laws of total expectation (LTE) and variance (LTV), which say
We've found that and
, so that by the LTE,
Next, we have
where the second moment is computed via
In turn, this gives