📊honors statistics review

key term - F(x)

Definition

F(x) is a mathematical function that represents the probability distribution function (PDF) for a discrete random variable. It is a fundamental concept in probability theory and statistics, as it describes the likelihood of different outcomes or values that a random variable can take on.

5 Must Know Facts For Your Next Test

  1. The function F(x) represents the probability that a discrete random variable X takes on a value less than or equal to x.
  2. F(x) is a non-decreasing function, meaning that as x increases, the value of F(x) either increases or remains the same.
  3. The values of F(x) range from 0 to 1, with F(x) = 0 representing the probability that X is less than or equal to the smallest possible value, and F(x) = 1 representing the probability that X is less than or equal to the largest possible value.
  4. The probability mass function (PMF) of a discrete random variable X, denoted as P(X = x), is related to the cumulative distribution function (CDF) F(x) by the formula: F(x) = ∑_{y ≤ x} P(X = y).
  5. The probability that a discrete random variable X takes on a value between a and b, inclusive, is given by F(b) - F(a-1).

Review Questions

  • Explain the relationship between the probability mass function (PMF) and the cumulative distribution function (CDF) for a discrete random variable.
    • The probability mass function (PMF) and the cumulative distribution function (CDF) are closely related for a discrete random variable. The PMF, denoted as P(X = x), gives the probability that the random variable X takes on a specific value x. The CDF, denoted as F(x), represents the probability that the random variable X is less than or equal to a given value x. The relationship between the two is that the CDF is the sum of the PMF values up to the given value x, or mathematically: F(x) = ∑_{y ≤ x} P(X = y). This means that the CDF can be calculated by adding up the probabilities of all the values of the random variable that are less than or equal to the given value x.
  • Describe the properties of the cumulative distribution function (CDF) F(x) for a discrete random variable.
    • The cumulative distribution function (CDF) F(x) for a discrete random variable has several important properties: 1. F(x) is a non-decreasing function, meaning that as x increases, the value of F(x) either increases or remains the same. 2. The values of F(x) range from 0 to 1, with F(x) = 0 representing the probability that X is less than or equal to the smallest possible value, and F(x) = 1 representing the probability that X is less than or equal to the largest possible value. 3. The probability that a discrete random variable X takes on a value between a and b, inclusive, is given by F(b) - F(a-1). 4. The relationship between the CDF F(x) and the probability mass function (PMF) P(X = x) is given by the formula: F(x) = ∑_{y ≤ x} P(X = y).
  • Explain how the cumulative distribution function (CDF) F(x) can be used to calculate probabilities for a discrete random variable.
    • The cumulative distribution function (CDF) F(x) can be used to calculate probabilities for a discrete random variable in several ways: 1. The probability that a discrete random variable X is less than or equal to a specific value x is given directly by the CDF: P(X ≤ x) = F(x). 2. The probability that a discrete random variable X is strictly less than a specific value x is given by: P(X < x) = F(x-1). 3. The probability that a discrete random variable X takes on a value between a and b, inclusive, is calculated as: P(a ≤ X ≤ b) = F(b) - F(a-1). 4. The probability mass function (PMF) P(X = x) can be obtained from the CDF by the formula: P(X = x) = F(x) - F(x-1). By using the properties and relationships of the CDF, you can easily calculate various probabilities related to a discrete random variable.