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Discrete Random Variables Solved Problems for Beginning Statistics

By — McGraw-Hill Professional
Updated on Aug 12, 2011

Review the following concepts if necessary:

Random Variable

  1. Let X represent the number of boys in families having three children. List all possible birth order permutations for families having three children and give the value of X for each outcome.
  2. Ans.   Table 5.11 gives the outcomes and the value X assigns to each outcome.

  3. For the experiment of rolling three dice, X is defined to be the sum of the three dice. What are the unique values assumed by X?
  4. Ans.   The values for X range from 3, corresponding to the outcome (1, 1, 1) to 18, corresponding to the outcome (6, 6, 6). The unique values are 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18.

Discrete Random Variable

  1. A telemarketing company administers an aptitude test consisting of 25 problems to potential employees. A variable of interest to the company is X, the number of problems worked correctly. How many different values are possible for X?
  2. Ans.   26, since an individual can get from 0 to 25 correct.

  3. A random variable assigns a single number to each outcome of an experiment. Is it true that each value of a random variable corresponds to a single outcome?
  4. Ans.   No. Consider the outcomes and values of X given in Table 5.11. The value X = 1 corresponds to the three outcomes BGG, GBG, and GGB for example.

Continuous Random Variable

  1. Identify the continuous random variables in parts (a) through (e).
    1. The time that an individual is logged onto the internet during a given week
    2. The number of domestic violence calls responded to per day by the Chicago police department
    3. The mortality rate for women who used estrogen therapy for at least a year starting in 1969
    4. The daily room rate for luxury/upscale hotels in the U.S.
    5. The number of executions per state since 1975
  2. Ans.   Parts (a) and (d) are continuous random variables. Time and money are almost always considered continuous even though in practice they are probably discrete.

  3. Is it possible to give a probability value to each individual value of a continuous random variable?
  4. Ans.   No. It is not possible to give an individual probability to each value of a continuous variable since the variable may assume an uncountably infinite number of different values. Instead, probabilities are assigned to an interval of values for a continuous random variable.

Probability Distribution

  1. According to the registrar's office at the University of Nebraska at Omaha (UNO), during the current semester, 9% of the students are registered for 3 credit hours, 13% are registered for 6 credit hours, 16% are registered for 9 credit hours, 21% are registered for 12 credit hours, 26% are registered for 15 credit hours, 13% are registered for 18 credit hours, and 2% are registered for 21 credit hours. If the random variable X represents the number of credit hours per student at UNO, give the probability distribution for X.
  2. Ans.   The probability distribution for X is given in Table 5.12.

  3. An experiment consists of rolling a die and flipping a coin. The coin has the number 1 stamped on one side and the number 2 stamped on the other side. The random variable Y is defined to equal the sum of the number showing on the coin plus the number showing on the die after the experiment is conducted. Give the probability distribution for Y.
  4. Ans.   Table 5.13 gives the outcomes for the experiment as well as the values the random variable Y assigns to the outcomes. From Table 5.13, the probability distribution given in Table 5.14 is determined.

    Y = 2 corresponds to one of twelve equally likely outcomes. Therefore P(2) equals = .083. Y = 3 corresponds to two of twelve equally likely outcomes. Therefore P(3) equals = .167. The other probabilities are found in a similar fashion.

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