Review Sheet for Test 2.

Here is a list of topics that we covered in class and some sample problems to look at.

• This section was on the Binomial Distribution. Some key points are:

  1. You begin with a sequence of Bernoulli trials. Each trial is independent and has the same success probability $p$.

  2. The formula for the probability distribution is easy to interpret: It is the number of ways to choose $k$ successes from $n$ times to succeed ($n$ choose $k$) times $p^k$ or one factor of $p$ for each success times $q^{n-k}$ or one $q$ for each failure.

  3. For small values of $n$ it is easy to compute exactly. For larger values, we noticed that the ratios are easier to deal with than the numbers them selves, as the numbers get really small.

  4. The most likely number of successes is int($np+p$). This is almost the mean $np$ but rounded in a fancy way up or down.

  5. The exact histogram for the binomial distribution appears bell shaped.

  Some sample problems from the Chapter review are: 1, 4, 6, 8.

• The Normal Approximation to the binomial. In this section we learned the following:

  1. What the family of normal curves are. They are the graphs of the function

     $$f_{\mu,\sigma}(x) = \frac{1}{(2\pi)^{1/2}\sigma} e^{-\left(\frac{x-\mu}{\sigma}\right)^2 }$$

     where $\mu$ is the mean and $\sigma$ is the standard deviation.

  2. We know that the graphs have a single maximum at $\mu$, and the inflection points are a distance $\sigma$ away from $\mu$.

  3. The area under the standard normal curve (when $\mu=0$ and $\sigma=1$) is given by an integral whose value is called $\Phi(z)$, and this is tabulated in the appendix.

  4. We learned how to use $\Phi(z)$ to find any area we desired.

  5. We know the areas of “$k$ standard deviations away from the mean” (68,95,99.8).

  6. We learned that if $S_n$ is the number of successes in $n$ trials then the probability that $a \leq S_n \leq b$ is approximately

     $$\Phi(\frac{b+1/2 - \mu}{\sigma}) - \Phi(\frac{a-1/2 -\mu}{\sigma})$$

     here $\mu = np$ and $\sigma = (npq)^{1/2}$. The extra 1/2’s are a nuisance and can be ignored if $\sigma$ is large enough.

  7. We learned about $\hat p$ the relative frequency for a sample and it closeness to $p$ the true probability. We derived in class the formula

     $$P(|\hat p - p| < \frac{k\sigma}{n}) \geq \Phi(k) - \Phi(-k)$$

     What did this mean about Al D’Amato losing the election?

  Some sample problems from the chapter review would be: 7, 10, 22

• The Poisson Approximation to the Binomial. If the Binomial histogram is really skewed, then the normal approximation is a bad one, and the Poisson approximation is better. The Poisson distribution is

  $$P( k \text{ succeses}) = \frac{\lambda^k}{k!}e^{-\lambda}.$$

  Some example problems would be: 13.

• Random sampling.

  1. We say several ways to count. Be sure you can give interpretations to “$n$ order $k$”, “$n!$”, “$n$ choose $k$”.

  2. Be clear about the two sample spaces: ordered sample space, vs. unordered sample spaces.

  3. Sampling with replacement. This gave us the binomial distribution. (Why?). For sampling with replacement only the ordered sample space has equally likely outcomes.

  4. Sampling without replacement. This gave us the hypergeometric distribution:

     $$\frac{\binom{G}{g}\binom{B}{b}}{\binom{G+B}{g+b}}$$

     We derived this two ways. By counting unordered samples, and by computing the ordered samples.

  5. Know how to count cards!

  Some sample problems would be: 12, 14, 17, 24

• This section is on Random Variables. For this test you will need to know how to find the distribution of a random variable. That is specify $P(X=k)$ for all possible $k$.
  Some examples would be from section 3.1 homework: 3, 6, 8, 9.