Questions and Answers
What are the two requirements for a discrete probability distribution?
The sum of the probabilities must equal 1, and each probability must be between 0 and 1, inclusive.
Determine whether the following random variable is discrete or continuous and state the possible values: (a) The number of points scored during a basketball game.
Discrete; possible values are x = 0, 1, 2, ...
Determine whether the following random variable is discrete or continuous and state the possible values: (b) The time it takes to fly from City A to City B.
Continuous; possible values are t > 0.
Determine whether the following random variable is discrete or continuous and state the possible values: (a) The number of people in a restaurant that has a capacity of 300.
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Determine whether the following random variable is discrete or continuous and state the possible values: (b) The distance a baseball travels in the air after being hit.
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Is the distribution a discrete probability distribution based on the following values: (0, 0.07), (1, 0.34), (2, 0.27), (3, 0.15), (4, 0.17)?
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Determine the required value of the missing probability P(4) to make the following distribution a discrete probability distribution: (3, 0.34), (4, ?), (5, 0.08), (6, 0.29).
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Is the following distribution a discrete probability distribution based on the values: (0, 0.263), (1, 0.576), (2, 0.127), (3, 0.029), (4, 0.004), (5, 0.001)?
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Describe the shape of the distribution represented by the random variable X for the number of marriages an individual aged 15 years or older has been involved in.
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Study Notes
Discrete Probability Distribution Requirements
- The sum of all probabilities must equal 1: ∑ P(x) = 1.
- Each probability must lie within the range 0 to 1, inclusive: 0 ≤ P(x) ≤ 1.
Random Variable Classification
- Random variables can be classified as discrete or continuous.
- Discrete Example: Number of points scored in a basketball game. Possible values: x = 0, 1, 2, ...
- Continuous Example: Time taken to fly between two cities. Possible values: t > 0.
Additional Random Variable Examples
- Discrete Example: Number of people in a restaurant with a capacity of 300. Possible values: x = 0, 1, 2, ..., 300.
- Continuous Example: Distance a baseball travels after being hit. Possible values: d > 0.
Validating Discrete Probability Distribution
- A given distribution of probabilities is valid if:
- The sum equals 1.
- All probabilities are between 0 and 1.
- Example probabilities:
- x = 0, P(x) = 0.07
- x = 1, P(x) = 0.34
- Sum = 1 confirms validity.
Finding Missing Probability
- In a probability distribution, if a value is missing, calculate it by ensuring the total equals 1.
- Example:
- Given P(3) = 0.34, P(5) = 0.08, P(6) = 0.29.
- Total given = 0.71, thus P(4) = 1.00 - 0.71 = 0.29.
Analyzing a Specific Probability Distribution
- Example distribution for marriages:
- x = 0, P(x) = 0.263
- x = 1, P(x) = 0.576
- x = 2, P(x) = 0.127
- x = 3, P(x) = 0.029
- x = 4, P(x) = 0.004
- x = 5, P(x) = 0.001
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Verification:
- All probabilities are between 0 and 1.
- Sum of probabilities = 1, confirming it’s a discrete probability distribution.
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Graph Characteristics:
- The distribution is unimodal (one mode) and skewed right.
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Description
Test your knowledge on discrete probability distributions and random variable classifications. This quiz explores the requirements for valid distributions and provides examples to help you identify discrete and continuous variables. Perfect for statistics students looking to enhance their understanding.
