By the end of this chapter, the student should be able to:
- Recognize and understand discrete probability distribution functions, in general.
- Calculate and interpret expected value.
- Recognize the binomial probability distribution and apply it
appropriately.
- Recognize the Poisson probability distribution and apply it
appropriately (optional).
- Recognize the geometric probability distribution and apply it
appropriately (optional).
- Recognize the hypergeometric probability distribution and apply it
appropriately (optional).
- Classify discrete word problems by their distributions.
A student takes a 10 question true-false quiz. Because the student had such a busy schedule, he or she could not study and randomly guesses at each answer. What is the probability of the student passing the test with at least a 70%?
Small companies might be interested in the number of long distance phone calls their employees make during the peak time of the day. Suppose the average is 20 calls. What is the probability that the employees make more than 20 long distance phone calls during the peak time?
These two examples illustrate two different types of probability problems involving discrete random variables. Recall that discrete data is data that you can count. A random variable describes the outcomes of a statistical experiment both in words and numerically. The values of a random variable can vary with each repetition of an experiment.
In this chapter, you will study probability problems involving discrete random distributions. You will also study long-term averages associated with them.
Upper case letters like XX or YY denote a random variable. Lower case letters like xx or yy denote the value of a random variable. If XX is a random variable, then XX is defined in words.
For example, let XX = the number of heads you get when you toss three fair coins. The sample space for the toss of three fair coins is
• TTTTTT; • THHTHH; • HTHHTH; • HHT HHT; • HTTHTT; • THTTHT; • TTHTTH; • HHH HHH. Then, xx = 0, 1, 2, 3. XX is in words and xx is a number. Notice that for this example, the xx values are countable outcomes. Because you can count the possible values that XX can take on (the xx values 0, 1, 2, 3), XX is a discrete random variable.
Toss a coin 10 times and record the number of heads. After all members of the class have completed the experiment (tossed a coin 10 times and counted the number of heads), fill in the chart using a heading like the one below. Let XX = the number of heads in 10 tosses of the coin.
Table 1
| XX |
Frequency of XX |
Relative Frequency of XX |
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- Which value(s) of XX occurred most frequently?
- If you tossed the coin 1,000 times, what values would XX take on? Which value(s) of XX do you think would occur most frequently?
- What does the relative frequency column sum to?
- Random Variable (RV):
- Variable (Random Variable):
A characteristic of interest in a population being studied. Common notation for variables are upper case Latin letters
XX size 12{X} {},
YY size 12{Y} {},
ZZ size 12{Z} {},...; common notation for specific value from the domain (set of all possible values of a variable) are lower case Latin letters
xx size 12{x} {},
yy size 12{y} {},
zz size 12{z} {},.... For example, if
XX size 12{X} {} is a number of children in a family, then domain is and
xx size 12{x} {} represents any integer from 0 to 20. Variable in statistics differs from variable in intermediate algebra in two following ways.
- The domain of random variable (RV) is not necessarily numerical set; it can be some “wording” set; for example, if
XX size 12{X} {} = hair color then the domain is {black, blond, gray, green, orange}.
- We can tell what specific value of
xx size 12{x} {} does the variable
XX size 12{X} {} take only after performing the experiment.
Before the experiment any value from domain is possible. For example, without ultrasound we can not tell the gender of a baby that should be delivered, but after delivery the gender is evident. More exact, every value from the domain is accompanied with some number
pp size 12{p} {},
0≤p≤10≤p≤1 size 12{0 <= p <= 1} {}, that characterizes the chance to have this value as an outcome of the experiment. In the example with gender,
p=12p=12 size 12{p= { {1} over {2} } } {}. That’s why statisticians use more exact name
“Random variable” (RV) instead of variable. Even more, they use word “distribution” having in the mind the RV, that is the pairing (value, probability of the value).
View the Video Lecture for this Chapter