Random Variables

 Main Concepts

We now make a slight shift from talking about events and outcomes to talking about random variables. Random variables are strange creatures. First, they are not variables in the algebraic sense. They are not "masking" a true value. Random variables have no values; instead, they have distributions.

Random variables are always numerical. Before we talked about the probability of selecting a red card, or a King. But random variables require numbers, and so you must declare "red" to be 1 (or 0, or 2, or any number you want.)

We're becoming a little more abstract in these probability units. Note that there's not much data around. We're no longer looking at data and making summaries.

• Frequently, in Statistics, the random experiment we care about most is the one in which a unit is randomly selected from a large population and some aspect of that unit is measured. The random variable represents the possible values we might see from our measurement, and the probability density function tells us the probability of seeing certain ranges of values.

•Just as we used mathematical functions to model the relative frequencies of values found within populations, we will now use probability density functions to model the probabilities associated with values in a random experiment.

•In the same sense that a probability is a long-term relative frequency, the Expected Value is a long-term average that is directly analogous to the population mean.

•The Law of Large Numbers tells us that as the sample size gets larger, sample averages approach Expected Values.

Important Notes about Densities

• The normal density is one of the most useful continuous pdfs.
Other continuous pdfs you might encounter are the t-distribution, the chi-square, the F, the continuous uniform.

•The binomial as a useful discrete pdf. Other discrete pdfs you might encounter are the geometric, the poisson, and the discrete uniform.