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Two Types of Random Variables

Random variables are sometimes hard for students to understand the nature of the random variable. Because it is a new idea, this may take lots of time to make sure students understand what exactly is going on. I have found that discreet random variables are slightly easier than continuous ones. The problem is that the random variable can take on a number of unknown values, which also tend to be represented as variables. Random variables can be thought of as bins, that hold any number of other variables. This is helpful when talking about distributions later on.  

Mean and Standard Deviation of Discrete Random Variables

This is where you need to connect back to the early lesson when we were pretty strict about calling the mean and the standard deviation parameters. When talking about distributions of a random variable, the most important thing is the mean and the standard deviation, and once the method of finding each is established most texts will then give a table of the common ones.

Up to this point there has been lots of theory. This section begins the statistical portion in earnest, and deserves some extra time for practice in finding means and variances. This is especially important before we start to define special distributions where means and variances will frequently be calculated by the rule for each individual distributions.  

The Binomial Distribution

This the big one for discreet distributions. The formula is one that needs to be memorize, as so many problems come down to the binomial distribution. The idea of a probability of pass or fail applies in many circumstances. One place that can be fun to look at for students (especially those who are sick of medical studies...) is to look at some of the probabilities that are published about sports. For example, currently Accuscore is famous for running many simulations of each game a team plays to come up with probabilities that each team will make it to the playoffs. Presented in popular media, the mechanics of what goes on behind the scenes is unknown. (Gambling disclaimer! Accuscore attempts to profit off of selling their information to people who use it to place bets. They technically run a monte carlo program with using a binomial distribution as the probability distribution function (my assumption, but I’d be really surprised if I’m wrong.) They try to bamboozle people out of their money with fancy language and guarantees, another reason knowledge is power.) Students are more than capable of running their own simulations, for sporting events or other types of contests. The trick that students will probably discover is that the binomial distribution is great, but there is a huge catch to it. How to find p.

I try to get my students to come up with the questions on their own. Every question they are likely to encounter has something along the lines of “given that p is .6 \ldots”, so when students are given a more open ended problem, hopefully they will ask how to find the probability of success. They can be guided to begin thinking about it; asking “Does this answer make sense? How do you think they chose p?” after solving some of the books problems. It isn’t always easy, or clear, how to choose success. Sometimes it is determined by a specific probability space, but for human studies, it can be a bit of a guessing game. It is an important consideration as students begin to learn about the job of statisticians and what is directly computed and what has to be assumed or chosen by the person doing the study.  

The Poisson Probability Distribution

The Poisson distribution, also sometimes called the exponential distribution, is very useful in practice. It is use frequently for risk management scenarios and other applied questions in physics and business. The topic will also be a good one to revisit when confidence intervals are studied, as the two frequently go hand in hand.

However, it is not a topic on the AP examination. In checking a number of textbooks for a first year probability and stats course at the university level, it seldom made an appearance in those as well. Therefore, how you use this section is dependent on what the goals of your class are, how skilled they are, and how badly you may need to speed things along before the examination. Again, it is a great topic, but may not fit with your plans for this course.  

The Geometric Probability Distribution

The geometric distribution has a very particular use. That is, how long before a single event will occur. The troubles students are likely to encounter are the geometric distribution’s similarity to the binomial distribution and the circumstances that it is applied.

The forms of the two distributions simply must be memorized. There unfortunately no way around it. Something to focus on is what the variable in each distribution represents. In the binomial distribution the variable is how many successes out of a fixed total. The geometric distribution’s variable is how many trials. Now if students are really clued in conceptually they will realize that the two can be confused because they are essentially the same. Look at a binomial probability of one success:

p(x)=np^1 (1-p)^{n-1}

The only difference is the n multiplied at the front. This is because the binomial distribution does not care when you have the success. The Geometric distribution only give the probability for a success at a particular point, i.e. the probability of 1 success in 5 trials as opposed to having the first success on the 5^{\mathrm{th}} trial.

Because of this choosing when to apply the geometric distribution is sometimes tricky. The thing to focus on is “Before the first success” or “until first success”. The number of successes is fixed in the geometric distribution, and that can be the clue pulled from the problem to choose correctly.  

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