Random Variables and their PDFs
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• We now make a slight shift from talking about events and outcomes to talking about random variables. Random variables are strange creatures. For one thing, they are not variables in the algebraic sense. They are not "masking" a true value. Instead, random variables have distributions.
• 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.
• Random variables are always numerical. Some students may incorrectly give "the color of a card dealt from a deck" as an example of a random variable. But a random variable would be "the number of red cards dealt from a deck of cards."
• 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's your pdf? 2) what's your expected value? 3) what's your standard deviation? and 4) what physical situations can you model?
•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?
• You might hear that P(X =x) = 0 for a continuous distribution. This means that, when the possible values are over an infinite spectrum, the probability of any one particular value occuring is 0. Remember that for a continuous distribution, the probability is given by the area under the curve, and the area under a single point is 0.
• 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.