Probability: Homework 9

This homework doesn’t need to be turned in. These are just practice problems to help learn the new things for the final.

Inequalities

  1. Flip a fair coin 100 times. Put an upper bound on the probability that at least 90 heads happened.

  2. Let \(X_1, \dots, X_n\) be iid random variables with mean \(\mu\) and standard deviation \(\sigma\).

    1. Apply Chebyshev’s inequality to \(P(|\bar{X}_n - \mu| \geq 2 \sigma)\). Hint. Your answer should have an \(n\) in it. Also, \(\sigma\) is not the standard deviation of \(\bar{X}_n\).

    2. Give a specific value of \(n\) such that \(\bar{X}_n\) is within \(2\sigma\) of the mean \(\mu\) with probability at least 99%.

    3. Imagine that you are talking to someone who doesn’t know probability, like a small child or a Supreme Court Justice. They ask, ‘If I have 100 people in my sample, is my sample average within 2 standard deviations of the true mean or not?’ What do you say to them?

  3. This exercise is about estimating the probability that a Poisson variable is exceptionally large using different inequalities. Let \(X \sim \text{Poisson}(\lambda)\).

  1. Is Markov’s inequality valid to use on \(X\)? If so, use it to estimate \(P(X \geq 2 \lambda)\).

  2. Explain why \(P(X \geq 2 \lambda) = P(e^X \geq e^{2 \lambda})\).

  3. Show that \[P(X \geq 2 \lambda) \leq e^{-\lambda (3 - e)}.\] Hint. Use Markov’s inequality on the previous part, and the mgf of the Poisson. (This is called Chernoff’s bound.)

  4. Which bound is better? Why?

Law of large numbers

  1. State the law of large numbers.

  2. Let \(X_1, X_2, \dots\) be iid, positive random variables with mean \(5\). Let \(Y_1, Y_2, \dots\) be iid, positive random variables with mean \(1\). Almost surely, what is the following limit? \[\lim_{n \to \infty} \frac{X_1 + \cdots + X_n}{Y_1 + \cdots + Y_n}\]

Central limit theorem

  1. State the central limit theorem.

  2. Let \(X_1, \dots, X_{100}\) be a sequence of 100 iid variables with mean \(10\) and standard deviation 5.

  1. Assuming that 100 is sufficiently large, what does the central limit theorem say about the sample mean \(\bar{X}\)?

  2. What, approximately, is the distribution of \(X_1 + \cdots + X_{100}\)?

  3. Estimate \(P(\bar{X} \leq 20)\) using \(\Phi\), the standard normal cdf.