{"id":147,"date":"2016-04-21T22:43:44","date_gmt":"2016-04-21T22:43:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstats1xmaster\/?post_type=chapter&p=147"},"modified":"2021-03-29T19:47:20","modified_gmt":"2021-03-29T19:47:20","slug":"introduction-discrete-random-variables-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-introstats1\/chapter\/introduction-discrete-random-variables-2\/","title":{"raw":"Introduction to Discrete Random Variables","rendered":"Introduction to Discrete Random Variables"},"content":{"raw":"
\r\n[caption align=\"center\" width=\"625\"]\"ThisYou can use probability and discrete random variables to calculate the likelihood of lightning striking the ground five times during a half-hour thunderstorm. (Credit: Leszek Leszczynski)[\/caption]\r\n<\/div>\r\n\r\n\r\n

A student takes a ten-question, true-false quiz. Because the student had such a busy schedule, he or she could not study and guesses randomly at each answer. What is the probability of the student passing the test with at least a [latex]70%[\/latex]?<\/p>\r\n

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 [latex]20[\/latex] calls. What is the probability that the employees make more than [latex]20[\/latex] long-distance phone calls during the peak time?<\/p>\r\n

These two examples illustrate two different types of probability problems involving discrete random variables. Recall that discrete data are data that you can count. A\u00a0random variable<\/strong> describes the outcomes of a statistical experiment in words. The values of a random variable can vary with each repetition of an experiment.<\/p>\r\n\r\n

Random Variable Notation<\/h1>\r\n

Upper case letters such as [latex]X[\/latex] or [latex]Y[\/latex] denote a random variable. Lower case letters like [latex]x[\/latex]\u00a0or [latex]y[\/latex] denote the value of a random variable. If [latex]X[\/latex] is a random variable, then [latex]X[\/latex] is written in words, and [latex]x[\/latex] is given as a number.<\/strong><\/p>\r\n\r\n

For example, let [latex]X[\/latex] = the number of heads you get when you toss three fair coins. The sample space for the toss of three fair coins is [latex]TTT[\/latex]; THH; HTH; HHT; HTT; THT; TTH; HHH[\/latex]. Then, [latex]x = 0, 1, 2, 3[\/latex]. [latex]X[\/latex] is in words and [latex]x[\/latex] is a number. Notice that for this example, the [latex]x[\/latex]values are countable outcomes. Because you can count the possible values that [latex]X[\/latex] can take on and the outcomes are random (the [latex]x[\/latex] values [latex]0, 1, 2, 3[\/latex]), [latex]X[\/latex] is a discrete random variable.<\/p>\r\n\r\n

\r\n
\r\n
\r\n

Activity<\/h3>\r\n
Toss a coin ten times and record the number of heads. After all members of the class have completed the experiment (tossed a coin ten times and counted the number of heads), fill in a table like the one below. Let [latex]X[\/latex] = the number of heads in ten tosses of the coin.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/th>\r\nFrequency of [latex]x[\/latex]<\/strong><\/th>\r\nRelative Frequency of [latex]x[\/latex]<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n
<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
    \r\n \t
  1. Which value(s) of [latex]x[\/latex] occurred most frequently?<\/li>\r\n \t
  2. If you tossed the coin [latex]1,000[\/latex] times, what values could [latex]x[\/latex]take on? Which value(s) of [latex]x[\/latex] do you think would occur most frequently?<\/li>\r\n \t
  3. What does the relative frequency column sum to?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n\r\n\r\n
    \r\n

    Glossary<\/h2>\r\n
    \r\n \t
    Random Variable (RV)<\/dt>\r\n \t
    a characteristic of interest in a population being studied; common notation for variables are upper case Latin letters [latex]X[\/latex], [latex]Y[\/latex], [latex]Z[\/latex],...; common notation for a specific value from the domain (set of all possible values of a variable) are lower case Latin letters [latex]x, y,[\/latex] and [latex]z[\/latex]. For example, if [latex]X[\/latex] is the number of children in a family, then [latex]x[\/latex] represents a specific integer [latex]0, 1, 2, 3,....[\/latex] Variables in statistics differ from variables in intermediate algebra in the two following ways.\r\n