Discrete Random Variables for Beginning Statistics
Practice problems for these concepts can be found at:
- Discrete Random Variables Solved Problems for Beginning Statistics
- Discrete Random Variables Supplementary Problems for Beginning Statistics
A random variable associates a numerical value with each outcome of an experiment. A random variable is defined mathematically as a real-valued function defined on a sample space, and is represented as a letter such as X or Y.
EXAMPLE 5.1 For the experiment of flipping a coin twice, the random variable X is defined to be the number of tails to appear when the experiment is performed. The random variable Y is defined to be the number of heads minus the number of tails when the experiment is conducted. Table 5.1 shows the outcomes and the numerical value each random variable assigns to the outcome. These are two of the many random variables possible for this experiment.
EXAMPLE 5.2 An experimental study involving diabetics measured the following random variables: fasting blood sugar, hemoglobin, blood pressure, and triglycerides. These random variables assign numerical values to each of the individuals in the study. The numerical values range over different intervals for the different random variables.
Discrete Random Variable
A random variable is a discrete random variable if it has either a finite number of values or infinitely many values that can be arranged in a sequence. We say that a discrete random variable may assume a countable number of values. Discrete random variables usually arise from an experiment that involves counting. The random variables given in Example 5.1 are discrete, since they have a finite number of different values. Both of the variables are associated with counting.
EXAMPLE 5.3 An experiment consists of observing 100 individuals who get a flu shot and counting the number X who have a reaction. The variable X may assume 101 different values from 0 to 100. Another experiment consists of counting the number of individuals W who get a flu shot until an individual gets a flu shot and has a reaction. The variable W may assume the values 1, 2, 3, . . .. The variable W can assume a countably infinite number of values.
Continuous Random Variable
A random variable is a continuous random variable if it is capable of assuming all the values in an interval or in several intervals. Because of the limited accuracy of measuring devices, no random variables are truly continuous. However, we may treat random variables abstractly as being continuous.
EXAMPLE 5.4 The following random variables are considered continuous random variables: survival time of cancer patients, the time between release from prison and conviction for another crime, the daily milk yield of Holstein cows, weight loss during a dietary routine, and the household incomes for single-parent households in a sociological study.
The probability distribution of a discrete random variable X is a list or table of the distinct numerical values of X and the probabilities associated with those values. The probability distribution is usually given in tabular form or in the form of an equation.
EXAMPLE 5.5 Table 5.2 lists the outcomes and the values of X, the sum of the up-turned faces for the experiment of rolling a pair of dice. EXCEL Table 5.2 is used to build the probability distribution of the random variable X. This table lists the 36 possible outcomes. Only one outcome gives a value of 2 for X. The probability that X = 2 is 1 divided by 36 or .028 when rounded to three decimal places. We write this as P(2) = .028. The probability that X = 3, P(3), is equal to 2 divided by 36 or .056. The probability distribution for X is given in Table 5.3.
Note that in Table 5.2 the expression =A2+B2 is entered into C2 and a click-and-drag is performed from C2 to C19 and the expression =E2+F2 is entered into G2 and a click-and-drag is performed from G2 to G19.
The probability distribution, P(x) = P(X = x) satisfies formulas (5.1) and (5.2).
- P(x) ≥ 0 for each value x of X (5.1)
- σ P(x) = 1 where the sum is over all values of X (5.2)
Notice that the values for P(x) in Table 5.3 are all positive, which satisfies formula (5.1), and that the sum equals 1 except for rounding errors.
EXAMPLE 5.6 P(x) = , x = 1, 2, 3, 4 is a probability distribution since P(1) = .1, P(2) = .2, P(3) = .3, and P(4) = .4 and (5.1) and (5.2) are both satisfied.
EXAMPLE 5.7 It is known from census data that for a particular income group that 10% of households have no children, 25% have one child, 50% have two children, 10% have three children, and 5% have four children. If X represents the number of children per household for this income group, then the probability distribution of X is given in Table 5.4.
The event X ≥ 2 is the event that a household in this income group has at least two children and means that X = 2, or X = 3, or X = 4. The probability that X ≥ 2 is given by
- P(X ≥ 2) = P(X = 2) + P(X = 3) + P(X = 4) = .50 + .10 + .05 = .65
The event X ≤ 1 is the event that a household in this income group has at most one child and is equivalent to X = 0, or X = 1. The probability that X ≤ 1 is given by
- P(X ≤ 1) = P(X = 0) + P(X = 1) = .10 + .25 = .35
The event 1 ≤ X ≤ 3 is the event that a household has between one and three children inclusive and is equivalent to X = 1, or X = 2, or X = 3. The probability that 1 ≤ X ≤ 3 is given by
- P(1 ≤ X ≤ 3) = P(X = 1) + P(X = 2) + P(X = 3) = .25 + .50 + .10 = .85
The above discussion may be summarized by stating that 65% of the households have at least two children, 35% have at most one child, and 85% have between one and three children inclusive.
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