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4.2 Probability Distribution of a Discrete Random Variable

LEARNING OBJECTIVES

  • Recognize, understand, and construct discrete probability distributions.

    The probability distribution for a random variable lists all the possible values of the random variable and the probability the random variable takes on each value.  The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable.  A probability distribution can be a table, with a column for the values of the random variable and another column for the corresponding probability, or a graph, like a histogram with the values of the random variable on the horizontal axis and the probabilities on the vertical axis.

    In a probability distribution, each probability is between 0 and 1, inclusive.  Because all possible values of the random variable are included in the probability distribution, the sum of the probabilities is 1.

    EXAMPLE

    A child psychologist is interested in the number of times a newborn baby’s crying wakes its mother after midnight.  For a random sample of 50 mothers, the following information was obtained. Let [latex]X[/latex] be the number of times per week a newborn baby’s crying wakes its mother after midnight.  For this example, the values of the random variable are [latex]x = 0, 1, 2, 3, 4, 5[/latex].

    In the table, the left column contains all of the possible values of the random variable and the right column, [latex]P(x)[/latex], is the probability that [latex]X[/latex] takes on the corresponding value [latex]x[/latex].  For example, in the first row, the value of the random variable is 0, and the probability the random variable is 0 is [latex]\displaystyle{\frac{2}{50}}[/latex].  In the context of this example, that means that the probability a newborn baby’s crying wakes its mother 0 times per week is [latex]\displaystyle{\frac{2}{50}}[/latex].

    [latex]x[/latex] [latex]P(x)[/latex]
    [latex]0[/latex] [latex]\frac{2}{50}[/latex]
    [latex]1[/latex] [latex]\frac{11}{50}[/latex]
    [latex]2[/latex] [latex]\frac{23}{50}[/latex]
    [latex]3[/latex] [latex]\frac{9}{50}[/latex]
    [latex]4[/latex] [latex]\frac{4}{50}[/latex]
    [latex]5[/latex] [latex]\frac{1}{50}[/latex]

    Because [latex]X[/latex] can only take on the values 0, 1, 2, 3, 4, and 5, [latex]X[/latex] is a discrete random variable.  Note that each probability is between 0 and 1, and the sum of the probabilities is 1:

     

    [latex]\displaystyle{\frac{2}{50}+\frac{11}{50}+\frac{23}{50}+\frac{9}{50}+\frac{4}{50}+\frac{1}{50}=1}[/latex]

    TRY IT

    Suppose Nancy has classes three days a week. She attends classes three days a week [latex]80\%[/latex] of the time, two days a week  [latex]15\%[/latex] of the time, one day a week [latex]4\%[/latex] of the time, and no days [latex]1\%[/latex]of the time. Suppose one week is randomly selected.

    1. Let [latex]X[/latex] be the number of days Nancy ____________________.
    2. [latex]X[/latex] takes on what values?
    3. Suppose one week is randomly chosen. Construct a probability distribution table like the one in the example above. The table should have two columns labelled [latex]x[/latex] and [latex]P(x)[/latex].  What does the [latex]P(x)[/latex] column sum to?
    Click to see Solution
    1. Let [latex]X[/latex] be the number of days Nancy attends class per week.
    2. 0, 1, 2, and 3.
    3. [latex]x[/latex] [latex]P(x)[/latex]
      [latex]0[/latex] [latex]0.01[/latex]
      [latex]1[/latex] [latex]0.04[/latex]
      [latex]2[/latex] [latex]0.15[/latex]
      [latex]3[/latex] [latex]0.80[/latex]

      The [latex]P(x)[/latex] column sums to 1.

    EXAMPLE

    Jeremiah has basketball practice two days a week. Ninety percent of the time, he attends both practices. Eight percent of the time, he attends one practice. Two percent of the time, he does not attend either practice. What is [latex]X[/latex], and what values does it take on?  Construct the probability distribution for this random variable.

    Solution

    [latex]X[/latex] is the number of days Jeremiah attends basketball practice per week and takes on the values 0, 1, and 2.

    [latex]x[/latex] [latex]P(x)[/latex]
    [latex]0[/latex] [latex]0.02[/latex]
    [latex]1[/latex] [latex]0.08[/latex]
    [latex]2[/latex] [latex]0.9[/latex]

    Video: “Random Variables and Probability Distributions” by Dr Nic’s Maths and Stats [4:39] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.

    Video: “Constructing a probability distribution for random variable | Khan Academy” by Khan Academy [6:47] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.

    Exercises

    1. A company wants to evaluate its attrition rate, in other words, how long new hires stay with the company. Over the years, they have established the following probability distribution. Let [latex]X[/latex] be the number of years a new hire will stay with the company.  Let [latex]P(x)[/latex] be the probability that a new hire will stay with the company [latex]x[/latex] years. So far, the company has created the following probability distribution table.
      [latex]x[/latex] [latex]P(x)[/latex]
      [latex]0[/latex] [latex]0.12[/latex]
      [latex]1[/latex] [latex]0.18[/latex]
      [latex]2[/latex] [latex]0.30[/latex]
      [latex]3[/latex] [latex]0.15[/latex]
      [latex]4[/latex]
      [latex]5[/latex] [latex]0.10[/latex]
      [latex]6[/latex] [latex]0.05[/latex]
      1. What number goes in the empty cell in the table?
      2. [latex]P(x = 1) =?[/latex]
      3. [latex]P(x \geq 5) =?[/latex]
      4. What does the column “[latex]P(x)[/latex]” sum to?
      Click to see Answer
      1. [latex]0.1[/latex]
      2. [latex]0.18[/latex]
      3. [latex]0.15[/latex]
      4. [latex]1[/latex]

       

    2. A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Let the random variable be the number of batches of muffins sold in the bakery.  Through observation, the baker has established a probability distribution.
      [latex]x[/latex] [latex]P(x)[/latex]
      [latex]1[/latex] [latex]0.15[/latex]
      [latex]2[/latex] [latex]0.35[/latex]
      [latex]3[/latex] [latex]0.40[/latex]
      [latex]4[/latex] [latex]0.10[/latex]
      1. What is the probability the baker will sell more than one batch?
      2. What is the probability the baker will sell exactly one batch?
      Click to see Answer
      1. [latex]0.85[/latex]
      2. [latex]0.15[/latex]

       

    3. Ellen has music practice three days a week. She practices for all of the three days [latex]85\%[/latex] of the time, two days [latex]8\%[/latex]of the time, one day [latex]4\%[/latex] of the time, and no days [latex]3\%[/latex] of the time. One week is selected at random.
      1. Define the random variable [latex]X[/latex].
      2. Construct a probability distribution table for the data.
      Click to see Answer
      1. The number of times Ellen practices each week.
      2. [latex]x[/latex] [latex]P(x)[/latex]
        [latex]0[/latex] [latex]0.03[/latex]
        [latex]1[/latex] [latex]0.04[/latex]
        [latex]2[/latex] [latex]0.08[/latex]
        [latex]3[/latex] [latex]0.85[/latex]

       

    4. Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events [latex]35\%[/latex] of the time, four events [latex]25\%[/latex] of the time, three events [latex]20\%[/latex] of the time, two events [latex]10\%[/latex] of the time, one event [latex]5\%[/latex] of the time, and no events [latex]5\%[/latex] of the time.
      1.  Define the random variable [latex]X[/latex].
      2. What values does [latex]x[/latex] take on?
      3. Construct the probability distribution table.
      4. Find the probability that Javier volunteers for less than three events each month.
      5. Find the probability that Javier volunteers for at least one event each month.
      Click to see Answer
      1. The number of times Javier volunteers each month.
      2. [latex]0, 1, 2, 3, 4, 5[/latex]
      3. [latex]x[/latex] [latex]P(x)[/latex]
        [latex]0[/latex] [latex]0.05[/latex]
        [latex]1[/latex] [latex]0.05[/latex]
        [latex]2[/latex] [latex]0.1[/latex]
        [latex]3[/latex] [latex]0.2[/latex]
        [latex]4[/latex] [latex]0.25[/latex]
        [latex]5[/latex] [latex]0.35[/latex]
      4. [latex]0.2[/latex]
      5. [latex]0.95[/latex]

       

    5. Suppose that the probability distribution for the number of years it takes to earn a Bachelor of Science degree is given in the following table.
      [latex]x[/latex] [latex]P(x)[/latex]
      [latex]3[/latex] [latex]0.05[/latex]
      [latex]4[/latex] [latex]0.40[/latex]
      [latex]5[/latex] [latex]0.30[/latex]
      [latex]6[/latex] [latex]0.15[/latex]
      [latex]7[/latex] [latex]0.10[/latex]
      1. In words, define the random variable [latex]X[/latex].
      2. What does it mean that the values zero, one, and two are not included for [latex]x[/latex] in the probability distribution?
      Click to see Answer
      1. The number of years it takes to earn a Bachelor of Science degree.
      2. The probabilities associated with those values is [latex]0[/latex].

       

    6. A physics professor wants to know what percent of physics majors will spend the next several years doing post-graduate research. He has the following probability distribution.
      [latex]x[/latex] [latex]P(x)[/latex]
      [latex]1[/latex] [latex]0.35[/latex]
      [latex]2[/latex] [latex]0.20[/latex]
      [latex]3[/latex] [latex]0.15[/latex]
      [latex]4[/latex]
      [latex]5[/latex] [latex]0.10[/latex]
      [latex]6[/latex] [latex]0.05[/latex]
      1. Define the random variable X.
      2. Define [latex]P(x)[/latex], or the probability of [latex]x[/latex].
      3. Find the probability that a physics major will do post-graduate research for four years.
      4. Find the probability that a physics major will do post-graduate research for at most three years.
      Click to see Answer
      1. The number of years a physics major spends doing post-graduate research.
      2. The probability that a physics major spends exactly [latex]x[/latex] years doing post-graduate research.
      3. [latex]0.15[/latex]
      4. [latex]0.7[/latex]

       

    7. A ballet instructor is interested in knowing what percent of each year’s class will continue on to the next, so that she can plan what classes to offer. Over the years, she has established the following probability distribution.
      [latex]x[/latex] [latex]P(x)[/latex]
      [latex]1[/latex] [latex]0.1[/latex]
      [latex]2[/latex] [latex]0.05[/latex]
      [latex]3[/latex] [latex]0.1[/latex]
      [latex]4[/latex]
      [latex]5[/latex] [latex]0.3[/latex]
      [latex]6[/latex] [latex]0.2[/latex]
      [latex]7[/latex] [latex]0.1[/latex]
      1. In words, define the random variable [latex]X[/latex].
      2. What number goes in the missing cell in the table?
      3. [latex]P(x \lt 4) = ?[/latex]
      4. What does the column [latex]P(x)[/latex] sum to and why?
      Click to see Answer
      1. The number of years a student will study ballet with the teacher.
      2. [latex]0.15[/latex]
      3. [latex]0.25[/latex]
      4. [latex]1[/latex] because all possible values of the random variable are included in the probability distribution.

       

    8. A theatre group holds a fund-raiser. It sells 100 raffle tickets for $5 apiece. Suppose you purchase four tickets. The prize is two passes to a Broadway show worth a total of $150.
      1. What are you interested in here?
      2. In words, define the random variable [latex]X[/latex].
      3. List the values that [latex]X[/latex] may take on.
      4. Construct the probability distribution.
      Click to see Answer
        1. How much money you will win or lose.
        2. The amount of money you spent/earned.
        3. [latex]-20, 130[/latex]
        4. [latex]x[/latex] [latex]P(x)[/latex]
          [latex]-20[/latex] [latex]0.96[/latex]
          [latex]130[/latex] [latex]0.04[/latex]

       

    9. Suppose that you are offered the following “deal.” You roll a die. If you roll a six, you win $10. If you roll a four or five, you win $5. If you roll a one, two, or three, you pay $6.
      1. What are you ultimately interested in here (the value of the roll or the money you win)?
      2. In words, define the random variable [latex]X[/latex].
      3. List the values that [latex]X[/latex] may take on.
      4. Construct the probability distribution.
      Click to see Answer
        1. How much money you will win or lose.
        2. The amount of money you win or lose.
        3. [latex]-6, 5, 10[/latex]
        4. [latex]x[/latex] [latex]P(x)[/latex]
          [latex]-6[/latex] [latex]0.5[/latex]
          [latex]5[/latex] [latex]0.3333[/latex]
          [latex]10[/latex] [latex]0.1667[/latex]

       

    10. People visiting video rental stores often rent more than one DVD at a time. The probability distribution for DVD rentals per customer at Video To Go is given in the following table. There is a five-video limit per customer at this store, so nobody ever rents more than five DVDs.
      [latex]x[/latex] [latex]P(x)[/latex]
      [latex]0[/latex] [latex]0.03[/latex]
      [latex]1[/latex] [latex]0.05[/latex]
      [latex]2[/latex] [latex]0.24[/latex]
      [latex]3[/latex]
      [latex]4[/latex] [latex]0.07[/latex]
      [latex]5[/latex] [latex]0.04[/latex]
      1. Describe the random variable [latex]X[/latex] in words.
      2. Find the probability that a customer rents three DVDs.
      3. Find the probability that a customer rents at least four DVDs.
      4. Find the probability that a customer rents at most two DVDs.
      Click to see Answer
      1. The number of DVDs a person rents at any one time.
      2. [latex]0.57[/latex]
      3. [latex]0.11[/latex]
      4. [latex]0.32[/latex]

       

    “4.2 Probability Distribution of a Discrete Random Variable” and “4.6 Exercises” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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