3.6 Extra Questions

  1. Suppose that \(\mathbb{E}[X] = 10\) and \(\mathrm{var}(X) = 2\). Also, suppose that \(Y=5 + 9 X\).

    1. What is \(\mathbb{E}[Y]\)?

    2. What is \(\mathrm{var}(Y)\)?

  2. Use the definition of variance to show that \(\mathrm{var}(bX) = b^2 \mathrm{var}(X)\) (where \(b\) is a constant and \(X\) is some random variable).

  3. Suppose you are interested in average height of students at UGA. Let \(Y\) denote a student’s height; also let \(X\) denote a binary variable that is equal to 1 if a student is female. Suppose that you know that \(\mathbb{E}[Y|X=1] = 5\' \ 4\"\) and that \(\mathbb{E}[Y|X=0] = 5\' \ 9\"\) (and that \(\mathrm{P}(X=1) = 0.5\)).

    1. What is \(\mathbb{E}[Y]\)?

    2. Explain how the answer to part (a) is related to the Law of Iterated Expectations.

  4. Consider a random variable \(X\) with support \(\mathcal{X} = \{2,7,13,21\}\). Suppose that it has the following pmf:

    \[ \begin{aligned} f_X(2) &= 0.5 \\ f_X(7) &= 0.25 \\ f_X(13) &= 0.15 \\ f_X(21) &= ?? \end{aligned} \]

    1. What is \(f_X(21)\)? How do you know?

    2. What is the expected value of \(X\)? [Show your calculation.]

    3. What is the variance of \(X\)? [Show your calculation.]

    4. Calculate \(F_X(x)\) for \(x=1\), \(x=7\), \(x=8\), and \(x=25\).

  5. What is the difference between consistency and unbiasedness?

  6. Suppose you have an estimator that is unbiased. Will it necessarily be consistent? If not, provide an example of an unbiased estimator that is not consistent.

  7. Suppose you have an estimator that is consistent. Will it necessarily be unbiased? If not, provide an example of a consistent estimator that is not unbiased.

  8. The Central Limit Theorem says that, \(\sqrt{n}\left(\frac{1}{n} \sum_{i=1}^n (Y_i - \mathbb{E}[Y])\right) \rightarrow N(0,V)\) as \(n \rightarrow \infty\) where \(V = \mathrm{var}(Y)\).

    1. What happens to \(n \left(\frac{1}{n} \sum_{i=1}^n (Y_i - \mathbb{E}[Y])\right)\) as \(n \rightarrow \infty\)? Explain.

    2. What happens to \(n^{1/3} \left(\frac{1}{n} \sum_{i=1}^n (Y_i - \mathbb{E}[Y])\right)\) as \(n \rightarrow \infty\)? Explain.