Probability problems day 16

  1. Let \(X \sim \text{Binomial}(100, 1/2)\).
  1. Put an upper bound on the probability that \(P(X \geq 75)\) using Markov’s inequality.
  2. What is the standard deviation of \(X\)?
  3. What is an upper bound for \(P(40 \leq X \leq 60)\)?

  1. A survey of 100 Hanover residents is drawn, and their annual incomes are averaged. Assuming that 100 is a Big Number, what is an approximation for the probability that the sample mean is within two standard deviations of the true population mean?

  2. Let \(U_1, U_2, \dots, U_{12}\) be Unif(0, 1) variables. Let \[\bar{X} = \frac{U_1 + U_2 + \cdots + U_{12}}{12}\] be the sample mean of the \(U_i\).

  1. What is the expected value of each \(U_i\)? Variance?

  2. What does the central limit theorem say about \(\bar{X}\)?

  3. Give an approximate distribution of \(X = U_1 + U_2 + \cdots + U_{12}\), including the parameters. Hint. The sum is \(12 \bar{X}\).

  4. Approximate \(P(X > 7)\) in terms of \(\Phi\), the standard normal cdf.

  1. Let \(X_1, \dots, X_n\) be iid Poisson(20) variables, and \(Y_1, \dots, Y_n\) iid Binomial(30, 1/2) variables. Almost surely, what is the following limit?

\[\lim_{n \to \infty} \frac{X_1 + \cdots + X_n}{Y_1 + \cdots + Y_n}\]