Page 149 - Statistics for Dummies

Chapter 8: Random Variables and the Binomial Distribution
A discrete random variable is countably infinite if its possible values can be
specifically listed out but they have no specific end. For example, the number
of accidents occurring at a certain intersection over a 10-year period can take
on possible values: 0, 1, 2, . . . (you know they end somewhere but you can’t
say where, so you list them all).
Continuous random variables typically represent measurements, such as
time to complete a task (for example 1 minute 10 seconds, 1 minute 20 sec-
onds, and so on) or the weight of a newborn. What separates continuous
random variables from discrete ones is that they are uncountably infinite;
they have too many possible values to list out or to count and/or they can be
measured to a high level of precision (such as the level of smog in the air in
Los Angeles on a given day, measured in parts per million).
Examples of commonly used continuous random variables can be found in
Chapter 9 (the normal distribution) and Chapter 10 (the t-distribution).
Probability distributions 133
A discrete random variable X can take on a certain set of possible outcomes,
and each of those outcomes has a certain probability of occurring. The nota-
tion used for any specific outcome is a lowercase x. For example, say you roll
a die and look at the outcome. The random variable X is the outcome of the
die (which takes on possible values of 1, 2, . . . , 6). Now if you roll the die and
get a 1, that’s a specific outcome, so you write “x = 1.”
The probability of any specific outcome occurring is denoted p(x), which
you pronounce “p of x.” It signifies the probability that the random variable X
takes on a specific value, which you call “little x.” For example, to denote the
probability of getting a 1 on a die, you write p(1).
Statisticians use an uppercase X when they talk about random variables in
their general form; for example, “Let X be the outcome of the roll of a single
die.” They use lowercase x when they talk about specific outcomes of the
random variable, like x = 1 or x = 2.
A list or function showing all possible values of a discrete random variable,
along with their probabilities, is called a probability distribution, p(x). For
example, when you roll a single die, the possible outcomes are 1, 2, 3, 4, 5,
1
and 6, and each has a probability of ⁄6 (if the die is fair). As another example,
suppose 40% of renters living in an apartment complex own one dog, 7% own
two dogs, 3% own three dogs, and 50% own zero dogs. For X = the number of
dogs owned, the probability distribution for X is shown in Table 8-1.

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