Question

3. Law of conditional variance. Let us define the conditional variance in a similar way to the conditional expectation. a. What is the object Var (Y|X = x) (i.e. is it a number, a random variable or a function)? What does it represent? b. Setting h(x) = Var (Y|X = x) we define the conditional variance as Var (Y|X) = h(X). What is this object? c. Prove the law of conditional variance: Var (Y) = E (Var (Y|X)) + Var (E (Y|X)) and describe it in words. d. We model the time at which a runner gets injured (in hours) during a marathon as an exponential random variable with parameter equal to 1 if the runner is under 30 years old and 2 if she is over 30. What is the mean and the standard deviation of the time at which a runner gets injured if 20% of the runners are over 30?

          3. Law of conditional variance. Let us define the conditional variance in a similar way to the conditional expectation.
a. What is the object Var (Y|X = x) (i.e. is it a number, a random variable or a function)? What does it represent?
b. Setting h(x) = Var (Y|X = x) we define the conditional variance as Var (Y|X) = h(X). What is this object?
c. Prove the law of conditional variance:
Var (Y) = E (Var (Y|X)) + Var (E (Y|X))
and describe it in words.
d. We model the time at which a runner gets injured (in hours) during a marathon as an exponential random variable with parameter equal to 1 if the runner is under 30 years old and 2 if she is over 30. What is the mean and the standard deviation of the time at which a runner gets injured if 20% of the runners are over 30?

3. Law of conditional variance. Let us define the conditional variance in a similar way to the conditional expectation.
a. What is the object Var (Y|X = x) (i.e. is it a number, a random variable or a function)? What does it represent?
b. Setting h(x) = Var (Y|X = x) we define the conditional variance as Var (Y|X) = h(X). What is this object?
c. Prove the law of conditional variance:
Var (Y) = E (Var (Y|X)) + Var (E (Y|X))
and describe it in words.
d. We model the time at which a runner gets injured (in hours) during a marathon as an exponential random variable with parameter equal to 1 if the runner is under 30 years old and 2 if she is over 30. What is the mean and the standard deviation of the time at which a runner gets injured if 20% of the runners are over 30?

Added by Daniel J.

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