5. Binomial Distribution & Discrete Random Variables
Discrete Random Variables
3:42 minutesProblem 4.T.5a
Textbook Question
The table shows the ages of students in a freshman orientation course.
a. Construct a probability distribution.
Verified step by step guidance1
Step 1: Understand the problem. A probability distribution lists all possible outcomes of a random variable along with their probabilities. Here, the random variable is the age of students, and the table provides the frequency of students for each age.
Step 2: Calculate the total number of students. Add up all the frequencies provided in the table: 2 (age 17) + 13 (age 18) + 4 (age 19) + 3 (age 20) + 2 (age 21) + 1 (age 22). This total will be used to calculate probabilities.
Step 3: Compute the probability for each age. Divide the frequency of students for each age by the total number of students. For example, the probability for age 17 is \( P(17) = \frac{2}{\text{Total Students}} \). Repeat this calculation for all ages.
Step 4: Construct the probability distribution table. Create a new table where each age is paired with its corresponding probability. Ensure that the sum of all probabilities equals 1, as this is a property of probability distributions.
Step 5: Verify your work. Double-check calculations to ensure accuracy and confirm that the probabilities sum to 1. This ensures the probability distribution is valid.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability Distribution
A probability distribution describes how the probabilities are distributed over the values of a random variable. In this context, it represents the likelihood of each age occurring among the students. To construct it, you divide the number of students at each age by the total number of students, resulting in a distribution that sums to 1.
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Relative Frequency
Relative frequency is the ratio of the number of times an event occurs to the total number of trials or observations. In the case of the ages of students, it is calculated by taking the count of students for each age and dividing it by the total number of students. This provides a way to express the probability of each age in the context of the overall group.
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Cumulative Frequency
Cumulative frequency is the running total of frequencies up to a certain point in a dataset. It helps in understanding the distribution of data by showing how many observations fall below a particular value. In this scenario, it can be useful to analyze how many students are aged 18 or younger, for example, by summing the frequencies of ages 17 and 18.
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