Have you ever played a game and tested the chances of random outcomes? This is similar to how we can describe and express the outcomes of random events with random variables.

Discrete random variables are variables with a finite number of potential outcomes. This means that the values are countable and limited within a range. For example, a set of books, a box of sugar cubes, a group of goats, and a person's shoe size are all examples of discrete random variables. In this lesson, we will delve into discrete random variables and their probability distributions in detail.

A discrete random variable is a variable that can only take on a limited number of specified values. Each value has a probability associated with it, and when all potential values are considered, the sum of probabilities must equal 1.

To understand discrete random variables, we must first understand distributions. The probability distribution for a discrete random variable X includes all of its potential values and the likelihood of X taking on each value in one experiment. In simpler terms, the distribution of a discrete random variable shows the probabilities associated with its values. The two most common types of discrete random variables are the binomial random variable (with a binomial probability distribution) and the geometric random variable (with a geometric probability distribution).

In this article, we will focus on binomial and geometric random variables, as they are relevant in AP Statistics courses. We will not be discussing other types such as Bernoulli, Multinomial, Hypergeometric, and Poisson distributions.

As mentioned earlier, the probability distribution of a discrete random variable is a list of its potential values and their corresponding probabilities. For a discrete random variable X, the following two conditions must be met:

- Each probability P(x) must be between 0 and 1: 0 ≤ P(x) ≤ 1.
- The sum of all probabilities must not exceed 1: ∑ P(x) = 1.

Let's consider an example to better comprehend the concept of a probability distribution for a discrete random variable.

Suppose we flip a fair coin twice and want to find the probability of getting at least one head. Let X represent the number of heads observed. The possible values of X are 0, 1, and 2, with equal chances of occurring. Therefore, the sample space is {hh, ht, th, tt}, where "hh" represents two heads, "ht" represents one head and one tail, and so on.

The probability distribution for X would look like this:

X012P(x)0.250.500.25

The probability of getting at least one head (X ≥ 1) can be calculated by adding the probabilities of X = 1 and X = 2, which are mutually exclusive events. Therefore, P(X ≥ 1) = P(1) + P(2) = 0.50 + 0.25 = 0.75. In other words, there is a 75% chance of getting at least one head when flipping a coin twice.

A binomial random variable is a type of discrete random variable that measures the frequency of a specific outcome across a set number of trials. It is represented by a binomial distribution.

For a discrete random variable to also be a binomial random variable, it must meet these criteria:

- The number of trials is predetermined or fixed.
- The trials are independent, meaning that the results of one trial do not affect the results of another.

Binomial random variables are used to gauge the number of successes in a given number of trials. In simpler terms, they deal with outcomes that are either a "success" or a "failure". This type of result is referred to as a "binary" outcome.

The chances of a successful or failed trial are equal, making it a "fair" experiment.

If a random variable (X) is classified as binomial, it has two parameters: "n" representing the number of trials and "p" representing the probability of success of an event.

For instance, let's say we randomly sample 125 nurses from a large hospital where 57% are female. In this case, X represents the number of female nurses in the sample. It can be considered a binomial random variable with n = 125 and p = 0.57.

To calculate the probability of X, use the formula:**P = nCx * p^x * (1-p)^(n-x)** where,P = binomial probabilitynCx = number of combinationsp = probability of success on a single trialn = number of trials

Geometric random variables also involve discrete outcomes and are commonly used in industries such as finance for cost-benefit analysis. They share similar characteristics with binomial random variables, but the number of trials is not predetermined. Instead, the number of trials will depend on the number of consecutive failures before a success occurs.

For example, let X = 3 represent rolling a fair die and getting a 3 as the outcome. In this experiment, the number of times the die is rolled before achieving a 3 is counted. Each time a 3 is rolled, it is considered a "success," and any other outcome is a "failure." The probability of X = 3 is 1/6 for each roll, as there are six sides on a die.

The formula for calculating the probability of a geometric random variable is: **P = p * (1-p)^(x-1)** where, 0 < p < 1 and x = 1, 2, 3...

Let's say a representative from the National Theatre Marketing Division randomly selects people on a street until finding someone who attended the last movie show with a 0.20 probability of success. In this case, the number of people selected until finding that person, denoted by X, follows a geometric random variable. If we want to find the probability of selecting 4 people before finding the desired person, the formula would be: **P = 0.20 * (1-0.20)^(4-1) = 0.1024**

When working with discrete random variables, it is important to understand the concepts of mean, variance, and standard deviation. These measures help us better understand the average, spread, and dispersion of values in a discrete probability distribution.

The mean, also known as the expected value, represents the average of the values in a discrete random variable. It is calculated by multiplying each value by its corresponding probability and then summing up all the values. The formula for mean is:

**µ = E(X) = ∑ xP(x)**

Calculating the mean is straightforward. We multiply each value by its probability and then add them together, as shown in the following example:

**Find the mean of the discrete probability distribution below:**

- x = -2, 1, 2, 3.5
- P(x) = 0.2, 0.34, 0.54, 0.31

**Solution:**

Using the formula, we get: µ = (-2) * 0.2 + (1) * 0.34 + (2) * 0.54 + (3.5) * 0.31 = 0.46

Variance measures the spread of data in a discrete random variable. It is the weighted average of the squared deviations from the mean. The formula for variance is:

**Var(X) = ∑ (x - µ)^2 * P(x)**

Similar to variance, standard deviation also measures the dispersion of data. It is the square root of the variance and is calculated using the following formula:

**SD(X) = √Var(X)**

Let's apply these concepts to a discrete random variable with the following probability distribution:

- x = -1, 0, 1, 4
- P(x) = 0.2, 0.5, α, 0.1

Given the following information:

- α = 1 - (0.2 + 0.5 + 0.1) = 0.2
- P(0) = 0.5
- P(X > 0) = P(1) + P(4) = 0.2 + 0.1 = 0.3
- P(X ≥ 0) = P(0) + P(1) + P(4) = 0.5 + 0.2 + 0.1 = 0.8
- Mean (µ) = (-1) * 0.2 + (0) * 0.5 + (1) * 0.2 + (4) * 0.1 = 0.4
- Variance = 0.4
- Standard Deviation = √0.4 = 0.632

- Discrete random variables have specified or finite values within an interval.Exploring Discrete Random Variables and Their Probability Distributions
- When it comes to random variables, there are two main categories: discrete and continuous. In this article, we will focus on discrete random variables and their corresponding probability distributions.
- Examples of Discrete Random Variables
- Discrete random variables are characterized by a countable, finite, or infinite set of possible values. Some common examples include the number of books in a pack, cubes of sugar in a box, and the number of goats in a pen. These variables can take on specific, distinct values and are not continuous.
- Understanding Probability Distributions
- A probability distribution for a discrete random variable is a comprehensive set that includes all possible values and their corresponding likelihood. This allows us to understand the likelihood of a particular outcome occurring.
- Binomial Random Variable Probability Distribution
- A binomial random variable is a type of discrete variable that follows a specific probability distribution. The formula for calculating the probability of a binomial random variable is P(X = k) = (n choose k) * p^k * (1-p)^(n-k). This distribution is useful in situations where there are only two possible outcomes, such as success or failure.
- Geometric Random Variable Probability Distribution
- A geometric random variable is another type of discrete variable with its own probability distribution. The formula for this distribution is P(X = k) = p * (1-p)^k-1, where k = 1, 2, 3, ... This distribution is often used to model the number of trials needed to achieve a success in a series of independent events.
- The Importance of Mean, Variance, and Standard Deviation
- To truly understand the behavior and characteristics of discrete random variables, it is essential to grasp the concepts of mean, variance, and standard deviation. These statistical measures provide valuable insights that can aid in analyzing data and conducting experiments.
- In conclusion, discrete random variables and their associated probability distributions play a crucial role in probability and statistics. By understanding and utilizing these concepts, we can make more informed decisions and gain a deeper understanding of the world around us.

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