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Questions and Answers
What is the definition of a random variable?
What is the definition of a random variable?
- A variable that can be measured and counted directly without randomness.
- A variable that has multiple categorical outcomes without numerical representation.
- A variable that has a single numerical value determined by chance for each outcome. (correct)
- A variable whose value is always the same in every trial.
Which characteristic is true for a probability distribution?
Which characteristic is true for a probability distribution?
- Each probability must be a whole number.
- The probabilities can be greater than 1, but must average to 1.
- The probabilities can sum to 0 as long as one probability is greater than 0.
- The sum of all probabilities must equal 1. (correct)
How do you calculate the expected value of a discrete random variable?
How do you calculate the expected value of a discrete random variable?
- By multiplying each outcome by its probability and summing the results. (correct)
- By selecting the outcome with the highest probability.
- By taking the average of all the possible outcomes.
- By summing the squares of the outcomes and dividing by the number of outcomes.
In a graph of a probability distribution, which of the following is typically represented on the x-axis?
In a graph of a probability distribution, which of the following is typically represented on the x-axis?
Which of the following statements is true regarding continuous random variables?
Which of the following statements is true regarding continuous random variables?
Which type of random variable is characterized by specific, countable outcomes?
Which type of random variable is characterized by specific, countable outcomes?
When analyzing a probability distribution, values such as 0.999 or 1.001 can occur due to what reason?
When analyzing a probability distribution, values such as 0.999 or 1.001 can occur due to what reason?
What distinguishes a continuous random variable from a discrete one?
What distinguishes a continuous random variable from a discrete one?
What is the range of valid probability values for a probability distribution?
What is the range of valid probability values for a probability distribution?
Which of the following is an example of a discrete random variable?
Which of the following is an example of a discrete random variable?
Given the probability distribution values, what is the issue with the following set: P(3) = 0.481, P(4) = 0.487, P(5) = -0.010?
Given the probability distribution values, what is the issue with the following set: P(3) = 0.481, P(4) = 0.487, P(5) = -0.010?
What characteristic differentiates a continuous random variable from a discrete random variable?
What characteristic differentiates a continuous random variable from a discrete random variable?
What would be the expected value of a game where you earn 2 points for heads and even, 1 point for tails and odd, and -1 for any other outcome?
What would be the expected value of a game where you earn 2 points for heads and even, 1 point for tails and odd, and -1 for any other outcome?
In a probability histogram, what does the vertical scale represent?
In a probability histogram, what does the vertical scale represent?
Which scenario exemplifies a continuous random variable?
Which scenario exemplifies a continuous random variable?
What defines the expected value in the context of probability?
What defines the expected value in the context of probability?
What characterizes a discrete random variable?
What characterizes a discrete random variable?
Which formula is used to calculate the mean for a probability distribution?
Which formula is used to calculate the mean for a probability distribution?
In a probability distribution, what does variance measure?
In a probability distribution, what does variance measure?
When calculating the standard deviation, what two elements are required?
When calculating the standard deviation, what two elements are required?
If a random variable takes on values between 0 and 1, what type of random variable is it?
If a random variable takes on values between 0 and 1, what type of random variable is it?
What is the first step in calculating the mean of a discrete probability distribution?
What is the first step in calculating the mean of a discrete probability distribution?
How would you describe the relationship between the mean and variance in a probability distribution?
How would you describe the relationship between the mean and variance in a probability distribution?
What does the probability distribution of a random variable describe?
What does the probability distribution of a random variable describe?
Flashcards
Random Variable
Random Variable
A variable whose value is a numerical outcome of a random phenomenon.
Probability Distribution
Probability Distribution
A description of the probabilities for each value of a random variable.
Discrete Random Variable
Discrete Random Variable
A random variable that can only take on specific values.
Continuous Random Variable
Continuous Random Variable
A random variable that can take on any value within a given range.
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Mean of a Probability Distribution
Mean of a Probability Distribution
The expected value (average value) of the random variable.
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Variance of a Probability Distribution
Variance of a Probability Distribution
A measure of how spread out the probability distribution is.
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Standard Deviation
Standard Deviation
The square root of the variance.
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Unusual Outcome
Unusual Outcome
An outcome that is not likely to occur by chance.
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Probability Distribution
Probability Distribution
A table or function showing all possible outcomes of a random variable and their probabilities.
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Discrete Random Variable
Discrete Random Variable
A variable that can only take on specific, separate values. Think whole numbers.
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Continuous Random Variable
Continuous Random Variable
A variable that can take on any value within any range. Think measurements.
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Expected Value
Expected Value
The average outcome of a random variable if the experiment is repeated many times.
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Probability
Probability
A number between 0 and 1 that describes the likelihood of an event occurring.
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Valid Probability Distribution
Valid Probability Distribution
All probabilities add up to 1 (or 100%).
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Probability Distribution Mean
Probability Distribution Mean
The expected value (average) of a random variable in a probability distribution.
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Probability Distribution Variance
Probability Distribution Variance
Measures how spread out the probability distribution is around the mean.
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Probability Distribution Standard Deviation
Probability Distribution Standard Deviation
The square root of the variance; expresses spread in the same units as the random variable.
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Calculating Mean (µ)
Calculating Mean (µ)
Sum of each value (x) multiplied by its probability (P(x))
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Calculating Variance (σ²)
Calculating Variance (σ²)
The sum of the squared deviations from the mean weighted by probabilities.
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Rounding Rule for µ, σ, σ²
Rounding Rule for µ, σ, σ²
Round results with one more decimal place than the random variable (x).
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TI-83/84 Calculator for Mean/Standard Dev.
TI-83/84 Calculator for Mean/Standard Dev.
Enter x values into L1, corresponding probabilities into L2, use 1-Var Stats L1,L2 function.
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Discrete Random Variable
Discrete Random Variable
A variable that can have only certain distinct values, often whole numbers.
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Continuous Random Variable
Continuous Random Variable
A variable that can take on any value within an interval.
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Green Pod Example
Green Pod Example
Illustrates discrete random variables and probability distribution calculation.
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Use of Results (Green Pod Example)
Use of Results (Green Pod Example)
Mean = expected average green pods out of 5. Variance = how spread out the results are.
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Section 5.2 - Random Variables
- Random variables are variables that have a single numerical value determined by chance for each outcome of a procedure.
- Probability distributions describe the probability for each value of a random variable. They can be presented as graphs, tables, or formulas.
- Consider distinguishing between outcomes that are likely by chance versus those that are unusual and unlikely by chance.
- Key concepts include understanding random variables, distinguishing between discrete and continuous random variables, and calculating mean, variance, and standard deviation for probability distributions.
- Determining whether outcomes are likely to occur by chance or if they are unusual, in the sense that they are unlikely to occur by chance is critical.
Characteristics of a Probability Distribution
- The sum of all probabilities must equal 1. Values slightly above or below 1 are acceptable due to rounding errors.
- Each probability value must be between 0 and 1 inclusive.
Discrete vs. Continuous Random Variables
- Discrete random variables have a finite number of values or a countable number of values (e.g., the number of tennis balls missed, the number of hairstyles).
- Continuous random variables have infinitely many values, and those values can be associated with measurements on a continuous scale (e.g., amount of coffee in ounces).
Expected Value
- The mean is also known as the expected value.
- The expected value of a discrete random variable (E(X)) is computed by summing the product of each value of the variable (x) and its corresponding probability (P(x)). E(X) = Σ[x * P(x)].
- A positive expected value means you are earning points on average; a negative means you lose points; a zero means you break even.
Interpreting Results
- The range rule of thumb suggests that most values lie within two standard deviations of the mean.
- Unusual results lie outside these limits.
- Probabilities used to identify if results are unusual. A probability below 0.05 for a high or low result suggests an unusual result.
How to Use a Calculator (TI-83/84)
- Enter random variable values (x) in list L1.
- Enter corresponding probabilities (P(x)) in list L2.
- Use the calculator's statistical functions to compute the mean (μ), variance (σ²), and standard deviation (σ).
Homework Assignments
- Page numbers and specific problem numbers are provided.
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