9 Questions for Chapter 9

9.1 Warm-up

Exercise 9.1 An astronomer is going to make a sequence of measurements with common mean \(\mu\) and standard deviation \(\sigma=4\). The astronomer will report the average \(\bar{X}\) of the \(n\) measurements and wants to choose \(n\) so that \(\pr{|\bar{X} - \mu| > 1} \approx 0.05\). How large must \(n\) be?

Exercise 9.2 The random variables \(X_1\), \(X_2\), …, \(X_{10}\) are independent and uniformly distributed on the interval \([0, 1]\). Using the central limit theorem, approximate \(\pr{ \sum_{i=1}^{10} X_i > 7}\).

Exercise 9.3 In an experiment to test for extrasensory perception, Fred tosses a coin repeatedly and Wilma, in the next room, tries to guess whether the coin falls heads or tails. Assuming that Wilma has a probability of 1/2 of correctly guessing each toss, find approximately the probability that she guesses correctly on at least 55% of the tosses, in a trial of 208 tosses of the coin.

Exercise 9.4 Let \(X\sim\mathcal{E}(\beta)\). Show that \(M_X(t) = \beta/(\beta - t)\), for \(t < \beta\). Use \(M_X\) to check that \(\expec{X} = 1/\beta\) and \(\var{X} = 1/\beta^2\).

Exercise 9.5 Let \(X \sim \text{Bin}(n,p)\). Show, directly or by expressing \(X\) as a sum, that \(M_X(t) = (pe^t +(1-p))^n\). Use \(M_X\) to check that \(\expec{X} = np\) and \(\var{X} = np(1-p)\).

Exercise 9.6 Let \(X\sim\text{Po}(\lambda)\) and \(Y \sim \text{Po}(\mu)\) be independent. Using moment generating functions, show that \(X+Y \sim \text{Po}(\lambda+\mu)\).

9.2 Work out

Exercise 9.7 Prove that the weak law of large numbers still holds whenever there are \(\mu\) and \(\sigma^2\) such that for all \(i\in\mathbb{N}\): \[\begin{aligned} \mu&=\expec{X_i}, & \sigma^2&\ge\var{X_i}. \end{aligned}\] In other words, prove that the weak law does not actually require the \(\var{X_i}\) to be equal; it is sufficient that the \(\var{X_i}\) have some upper bound.

Exercise 9.8 Let \(T\) be the total score obtained by rolling 10 fair dice. Use the central limit theorem to approximate \(\pr{T = 36}\) and \(\pr{30 \leq T \leq 40}\). hint:Use a continuity correction, for instance for \(\{T = 36\}\), use \(\{35.5 < X < 36.5\}\) when \(X\) is a continuously distributed random variable that approximates \(T\).

Exercise 9.9 A manufacturing process is designed to produce bolts with a 0.5cm diameter. Once a day, a random sample of 36 bolts is selected and the diameters recorded. If the average of the 36 values is less than 0.49cm or greater than 0.51cm, then the process is shut down for inspection and adjustment. The standard deviation for individual diameters is 0.02cm.

Find approximately the probability that the line will be shut down unnecessarily (i.e. if the true process mean really is 0.5cm).

9.3 Stretch

Exercise 9.10 Suppose \(X_1\), \(X_2, ...\) are Bernoulli random variables all with parameter \(q\), i.e. \(X_i\sim\text{Bin}(1,q)\). Let \(N\) be a random variable taking values in \(\mathbb{N}\). Suppose that the \(X_i\) are independent conditionally on \(N=n\), for all \(n\in\mathbb{N}\). Let \(Y =\sum_{i=1}^N X_i\) (where \(Y = 0\) if \(N = 0\)). Show that \[M_Y(t)= M_N\bigl( \log (qe^t + 1 - q) \bigr).\] Apply this to the situation of Exercise 6.30 where \(N \sim \text{Po}(\lambda)\) to confirm that \(Y \sim \text{Po}(\lambda q)\).

Exercise 9.11 Consider a random variable \(X\) taking non-negative integer values. The probability generating function of \(X\) is \(G_X(t) := \expec { t^X }\) for \(t \in \mathbb{R}\). Let \(E\) denote the event that \(X\) is even.

Given an explicit expression for the indicator random variable \(𝟙\{E\}\) as a function of \(X\), making use of the function \((-1)^X\). Hence express \(\pr{E}\) in terms of \(G_X(t)\).
Give \(\pr{E}\) when \(X \sim \text{Bin}(n,p)\); compare with Exercise 6.21.
Give \(\pr{E}\) when \(X \sim \text{Po} (\lambda)\).

Exercise 9.12 Prove the central limit theorem using M1, M3, M4, and M5. Use the following facts:

If \(Z\sim\mathcal{N}(0,1)\) then \(M_Z(t)=e^{t^2/2}\) (see Exercise 8.29).
If \(\lim_{n\to\infty}a_n=a\) then \(\lim_{n\to \infty}\left( 1 + \frac{a_n}{n} \right)^n=e^a\).
If \(m\) is twice differentiable, then there is a function \(h\) with \(\lim_{u\to 0}h(u)=0\) such that \[m(u) = m(0) + u m'(0) + \frac{u^2}{2!}m''(0) + u^2 h(u).\]