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: $x<em>1, x</em>2, x<em>3, …, x</em>k$

  • Probability: $p<em>1, p</em>2, p<em>3, …, p</em>k$

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:
    $ext{σ}^2<em>X = E[(X - ext{µ}</em>X)^2]$

• 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: $ext{σ}^2<em>X = ext{Sum of }((x - ext{µ}</em>X)^2 imes p(x))$

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