# 14.7. Exercises¶

1. A non-negative integer valued random variable $$X$$ has probability generating function given by $$G(s) = 0.4s + 0.3s^3 + 0.3s^5$$ for all $$s$$.

(a) Find the distribution of $$X$$.

(b) Find the probability generating function of $$2X$$.

(c) Find the probability generating function of $$X+Y$$ where $$Y$$ is independent of $$X$$ and has the same distribution as $$X$$.

(d) Use the result of Part (c) to find the distribution of $$X+Y$$.

2. My friend and I gamble on rolls of a die. Each time the die is rolled,

• my friend gives me a dollar if the number of spots is five or six,

• I give my friend a dollar if the number of spots is one or two,

• and no money changes hands if the number of spots is three or four.

If we play this game $$400$$ times, approximately what is the chance that my net gain is more than $$20$$ dollars?

3. Sketch the rough shape of the distribution of the proportion of heads in $$400$$ tosses of a coin. Find the numerical values of the expectation and the SD and mark them appropriately on your sketch.

4. Sketch the rough shape of the Poisson $$(625)$$ distribution and explain your choice. Find the numerical values of the expectation and SD and mark them appropriately on your sketch.

5. Let $$A_n$$ be the average of an i.i.d. random sample of size $$n$$ drawn from a population that has mean $$100$$ and SD $$2$$.

(a) What can you say about the value of $$P(A_n \text{ is in the interval } 100 \pm 2)$$ if $$n = 4$$?

(b) What can you say about the value of $$P(A_n \text{ is in the interval } 100 \pm 0.2)$$ if $$n = 400$$?

6. In the old days, the SD of the General Math SAT scores remained quite steady at around $$100$$ points every year while the mean fluctuated. In such a situation, how large should an i.i.d. sample of scores be so that you can be about $$95\%$$ confident of estimating the population mean score in a particular year correct to within $$1$$ point?

7. Ages in a large population have an average of $$30$$ years and a standard deviation of $$20$$ years. Two people are sampled randomly from the population.

(a) Could the distribution of ages in the population be roughly normal? Explain.

(b) Explain why the ages of the two sampled people are essentially independent of each other.

(c) Find upper and lower bounds for the probability that the ages of the two sampled people differ by at least $$60$$ years.

8. Use probability theory to show that as $$n \to \infty$$,

$e^{-n} \big{(} 1 + \frac{n}{1!} + \frac{n^2}{2!} + \cdots + \frac{n^n}{n!} \big{)} ~ \to ~ \frac{1}{2}$