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QUAN 2010 Notes Introduction to Business Statistics

Lauren M. Nelsen and Kimberly Swetz

Section 5.1 Random Variables

Many things in our world involve some level of uncertainty. We want to use tools from probability to make decisions and see what we can confidently say about whatever it is we’re investigating.
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A random variable is a variable whose possible values are the result of some random process, or experiment.
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Examples of Random Variables:
  • Coin toss: The number of heads from tossing two coins
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  • Dice: The outcome of rolling a die
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  • Stock dividend: The expected dividend payment on a stock
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  • Investment returns: The expected standard deviation of investment returns
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  • Cars passing a road: The number of cars that drive through a certain intersection in a given time period
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Definition 5.1.1.

Two Types of Random Variables:
  • discrete random variable: outcomes are whole numbers after performing an experiment
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  • continuous random variable: outcomes are any numerical value (including decimals or fractions) after performing an experiment
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Exercise 5.1.2.

Which of the following are discrete random variables?
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  • The number of heads from tossing two coins
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  • The outcome of rolling a die
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  • The expected divident payment on a stock
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  • The expected standard deviation of investment returns
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  • The number of cars that drive through a certain intersection in a given time period
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  • The height of a random student at UCCS
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Exercise 5.1.3.

Which of the following are continuous random variables?
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  • The number of heads from tossing two coins
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  • The outcome of rolling a die
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  • The expected divident payment on a stock
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  • The expected standard deviation of investment returns
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  • The number of cars that drive through a certain intersection in a given time period
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  • The height of a random student at UCCS
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Random variables can be “distributed” differently. For example, think about how rolls of a die would be distributed versus how the heights of UCCS students would be distributed.
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We’re going to start by discussing distributions of discrete random variables, and then in Chapter 7 we will discuss distributions of continuous random variables.
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Definition 5.1.4.

A discrete probability distribution is a listing of all possible outcomes of an experiment for a discrete random variable along with the relative frequency of each outcome
  1. Each outcome in the distribution must be mutually exclusive
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  2. \(0\leq P(X)\leq 1\) for all \(X\text{;}\) i.e., the probability of each outcome is between 0 and 1
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  3. \(\sum P(X)=1\text{;}\) the sum of all probabilities is 1
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