Discrete Random Variable

Definition

A discrete random variable is a variable that can only take on a countable number of distinct values, usually integers. It represents a quantity that is measured or observed in a random experiment, where the outcome can only be one of a set of specific, non-overlapping values.

5 Must Know Facts For Your Next Test

1. The probability distribution function (PDF) for a discrete random variable gives the probability of each possible value the variable can take on.
2. The mean, or expected value, of a discrete random variable is calculated by multiplying each possible value by its corresponding probability and summing the results.
3. The standard deviation of a discrete random variable measures the spread or variability of the possible values around the mean.
4. The discrete distribution known as the 'Lucky Dice Experiment' involves rolling a fair six-sided die and observing the number that comes up, which is a discrete random variable.
5. Discrete random variables are often used to model count data, such as the number of defects in a product, the number of customers arriving at a store, or the number of goals scored in a soccer game.

Review Questions

• Explain how the probability distribution function (PDF) is used to describe a discrete random variable.
  • The probability distribution function (PDF) for a discrete random variable provides the probabilities of each possible value the variable can take on. The PDF specifies the likelihood of observing a particular value, such as the probability of rolling a 3 on a six-sided die. By understanding the PDF, you can determine the probability of any specific outcome or range of outcomes for the discrete random variable.
• Describe how the mean (expected value) and standard deviation are calculated and interpreted for a discrete random variable.
  • The mean, or expected value, of a discrete random variable is calculated by multiplying each possible value by its corresponding probability and then summing the results. This provides the average or typical value we would expect to observe. The standard deviation measures the spread or variability of the possible values around the mean. A larger standard deviation indicates more dispersion in the possible values, while a smaller standard deviation means the values are clustered more closely around the mean.
• Analyze how the 'Lucky Dice Experiment' demonstrates the characteristics of a discrete random variable.
  • The 'Lucky Dice Experiment' involves rolling a fair six-sided die and observing the number that comes up. This is an example of a discrete random variable because the possible outcomes (1, 2, 3, 4, 5, or 6) are a countable set of distinct, non-overlapping values. The probability distribution function (PDF) would describe the likelihood of each face value appearing, with each value having an equal probability of 1/6. The mean and standard deviation of the die roll can also be calculated to further characterize this discrete random variable.

Related terms

Probability Distribution Function (PDF): The probability distribution function (PDF) describes the probability of each possible value that a discrete random variable can take on.

Expected Value: The expected value, or mean, of a discrete random variable is the weighted average of all possible values, where the weights are the probabilities of each value occurring.

Discrete Distribution: A discrete distribution is a probability distribution where the random variable can only take on a countable number of distinct values, such as the binomial or Poisson distributions.