Unit 7: Random Variables and their PDFs

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 Main Concepts

• We now make a slight shift from talking about events and outcomes to talking about random variables. Random variables are strange creatures, and students often are confused by the term as the words “random” and “variables” have many meanings. For one thing, they are not variables in the algebraic sense. They are not "masking" a true value. Instead, random variables have distributions. Random variables are functions that map events and outcomes to real numbers.

• 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.

• 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.

• For example, we previously used normal distributions to model relative frequencies of a continuous quantitative variable. In this context, we used the normal model to answer questions like “What percentage of the population of women is taller than 62 inches?” and the area under the normal curve represents a relative frequency. In this unit, normal probability density functions model the probability of seeing certain ranges of values in long-term sampling. In this context, we use the normal model to answer questions like “what is the probability that a women (or a sample of n women) is taller than 62 inches?”

• PDFs (Probability density functions) and random variables are joined at the hip. You can't mention one without, at least implicitly, mentioning the other. When you meet a random variable (at least in this course) for the first time, you should ask it 1) what is the shape of the graph of the pdf? 2) what is the center? and 3) what is the spread?

• Another question you might want to ask relating to random variables is: what physical situations can you model? Simulation is a very powerful tool in statistics, which you will learn more about in the next unit.

•For now, we assume the pdf is known and use this to make predictions about what we'll see in our observations. Later, we'll start with observations and go backwards: what can we infer about the pdf given the data we've seen?

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

•The binomial probability distribution is one of many useful discrete pdfs. Other discrete pdfs you might encounter are the geometric, the poisson, and the discrete uniform.

• In the same sense that a probability is a long-term relative frequency, the Expected Value of a random variable 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.