Discrete Random Variables and Binomial Distribution Study Notes

Discrete Random Variables and Binomial Distribution

Sections 5.3-5.4

Two Types of Random Variables

  • Random Variable: A variable that assumes numerical values associated with the outcomes of a random trial. Each outcome of an experiment has one and only one numerical value assigned.

    • Types:

    • Discrete

    • Continuous

Discrete Random Variable

  • Definition: A discrete random variable is one where possible values can be counted or listed.

    • Examples:

    • The number of defective units in a batch of 20.

    • A listener rating (on a scale of 1 to 5) in an AccuRating music survey.

Continuous Random Variable

  • Definition: A continuous random variable can assume any numerical value within one or more intervals.

    • Examples:

    • The waiting time for a credit card authorization.

    • The interest rate charged on a business loan.

Two Types of Quantitative Variables

Discrete and Continuous Variables

  • Discrete Variables: Counted values.

    • Example: Number of students in a class (you cannot have half a student).

    • Example: Possible results of rolling two dice: values can only be between 2 and 12 (i.e., 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12).

Continuous Variables

  • Definition: Continuous data can take any value within a range.

    • Examples:

    • A person's height (could be any value within the range of human heights).

    • Time in a race (could be measured to fractions of a second).

    • A dog's weight.

    • Length of a leaf.

Discrete Probability Distributions

  • Definition: The probability distribution of a discrete random variable is a table, graph, or formula that specifies the probability associated with each possible value of the variable.

    • Term: This is called a discrete probability distribution.

  • Notation: Denote the values of the random variable by x and the value's associated probability by p(x).

Requirements of a Discrete Probability Distribution

  1. Each probability must satisfy: 0 < p_i < 1 for all individual probabilities.

  2. The sum of all probabilities must equal 1:
    ext{Sum of } p(x) = 1 .

General Format of Discrete Probability Distribution

  • Table: A typical layout for the values of the random variable (X) and their probabilities is:

    • Value of X: x1, x2, x3, …, xk

    • Probability: p1, p2, p3, …, pk

Mean of a Discrete Random Variable

  • Definition: The mean of a set of observations is their arithmetic average.

  • Formula: The mean of a discrete random variable X is the weighted average of the possible values of X, using their corresponding probabilities as weights. This is denoted as:

    • Symbol: ext{µ}_X ext{ or } E(X)

  • Interpretation: The mean is also referred to as the expected value of X.

Calculation of the Mean

  • For a discrete random variable X with a probability distribution, the mean is calculated by: E(X) = ext{µ}_X = ext{Sum of } (x imes p(x))

    • Here, each possible value of X is multiplied by its corresponding probability, and the products are summed.

Variance and Standard Deviation of a Discrete Random Variable

  • Variance: A measure of the spread of a variable. It reflects how much the values of the variable differ from the mean.

    • Formula: The variance of a discrete random variable X is the weighted average of the squared deviations from the mean:
      ext{σ}^2X = E[(X - ext{µ}X)^2]

  • Interpretation: A larger variance indicates that the values of X are more spread out on average.

  • Standard Deviation: The square root of the variance gives the standard deviation of X:

    • Symbol: ext{σ}_X

Calculation of Variance

  • For a discrete random variable X with probability distribution and mean µX, the variance is computed by: ext{σ}^2X = ext{Sum of }((x - ext{µ}X)^2 imes p(x))

    • Each squared deviation is multiplied by the corresponding probability and summed.

The Binomial Distribution

  • Definition: A binomial experiment is characterized by the following features:

    • It consists of n identical trials.

    • Each trial results in either a “success” or a “failure.”

    • The probability of success, p, remains constant from trial to trial (the probability of failure is therefore 1 - p).

    • The trials are independent.

  • Random Variable: Let X represent the total number of successes in n trials of a binomial experiment; then X is a binomial random variable.

Binomial Distribution Formula

  • Note: The specific formula for calculating a binomial distribution will not be used directly; instead, Excel will be used for these computations.

  • Probability Expression for Binomial Random Variable: P(X = x) = p(x) = rac{n!}{x!(n−x)!} p^x(1−p)^{n−x}

    • Notation:

    • n! is read as "n factorial" and is computed as: n! = n imes (n-1) imes (n-2) imes … imes 1

    • Special case: 0! = 1

    • Factorial is not defined for negative numbers or fractions.

Mean and Variance of a Binomial Random Variable

  • For a binomial random variable X with parameters n and p:

    • Mean/Average/Expected Value:
      m_X = E(X) = np

    • Variance:
      s^2_X = np(1 - p)

    • Standard Deviation:
      ext{σ}_X = ext{sqrt}(np(1 - p))