A statistical experiment is any process by which measurements are obtained. Examples are
1. Counting the number of eggs in a robin’s nest
2. Counting the number of defective light bulbs in a case of bulbs
3. Measuring daily rainfall
4. Measuring the weight of a polar bear cub
Let x represent a quantitative variable that is measured or observed in an experiment. We are interested in the numerical values that x can take on. So x = number of eggs in a robin’s nest and x = weight in kilograms of a polar bear cub would be examples of such quantitative variables. Furthermore, we say that the quantitative variable x is a random variable because the value that x takes on in a given experiment is a chance or random outcome. We will study two types of random variables: discrete random variables and continuous random variables.
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Discrete random variable: When the observations of a quantitative random variable can take on only a finite number of values or a countable number of values, we say that the variable is a discrete random variable.
What is a countable number of values? As an example, let the random variable x be the number of wells that are drilled before the first productive well is found. Then x could be any of the values 1, 2, 3 …. In theory, we have an infinite number of possibilities for the values of x. The set of values of x corresponds to the set of counting numbers. Therefore, this type of infinity is called countable, and we say x has a countable number of values.
In most of the cases we will consider, a discrete random variable will be the result of a count. For instance, the number of students in a certain section of a statistics course this term is a discrete random variable. The value must be a counting number such as 14, or 20, or 34, and so forth. The values 25.34 or 25 1/2 are not possible.
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Continuous random variable: When the observations of a quantitative random variable can take on any of the countless number of values in a line interval, we say that the variable is a continuous random variable.
We will see most continuous random variables occurring as the result of a measurement. For example, the air pressure in an automobile tire represents a continuous random variable. The air pressure could in theory take on any value from 0 lb/in2 or psi to the bursting pressure of the tire. Values such as 28.10 psi, 28.12 psi, and so forth are possible. Another example is the height of students in your statistics class. The heights could take on any value from a low of 3 feet to a high of 7.5 feet.
In general, measurements of quantities such as length, weight, volume, temperature, or time yield continuous random variables. If the temperature changes from 72°F to 73°F, for example, it must take on all the temperature values between 72 and 73. Temperatures cannot just jump from one reading to the next. Discrete random variables often come from counts, such as the number of passing scores on an exam or the number of weeds in a garden.
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Examples: Which of the following random variables are discrete and which are continuous?
a. The time it takes a student selected at random to register for the fall term. Time can take on any value, so this is a continuous random variable.
b. The number of bad checks drawn on a bank on a day selected at random. The number of bad checks can be only a whole number such as 0, 1, 2, 3, etc. This is a discrete variable.
c. The amount of gasoline needed to drive your car 200 miles. We are measuring volume, which can assume any value, so this is a continuous random variable.
d. The number of voters pick in a random sample of 50 registered voters in a district in order to find the number who voted in the last county election. This is a count, so the variable is discrete.
