(Week 5 ) Probability Distributions: Random Variables
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Questions and Answers

Which of the following is an example of a discrete random variable?

  • The temperature in a room
  • The number of complaints per day (correct)
  • The weight of an object
  • The height of a student

    • A continuous random variable can assume only a finite number of values.

    False

    What is the expected value (E[x]) calculated from the frequency distribution?

  • 1.00
  • 1.20 (correct)
  • 0.75
  • 1.17

    • Define a random variable.

    <p>A random variable takes on different numerical values based on chance, arising from a random experiment.</p> Signup and view all the answers

    The standard deviation calculated from the fault frequency distribution is 1.20.

    <p>False</p> Signup and view all the answers

    The expected value of a random variable is denoted as E(____).

    <p>x</p> Signup and view all the answers

    What is the relative frequency of having 2 faults?

    <p>0.125</p> Signup and view all the answers

    Match the following types of outcomes with their descriptions:

    <p>Number of rings before the phone is answered = Finite number of values Game result: Won or Lost = Only two possible outcomes Height of trees in a forest = Uncountable infinite number of values Number of TVs in a household = Finite number of values</p> Signup and view all the answers

    What does the standard deviation measure in a dataset?

    <p>The spread or dispersion in a set of data</p> Signup and view all the answers

    The sum of the squared deviations multiplied by their probabilities in the standard deviation calculation is equal to _____

    <p>1.36</p> Signup and view all the answers

    The expected value is influenced by the probabilities of the outcome values.

    <p>True</p> Signup and view all the answers

    Match the following calculations with their results:

    <p>Expected Value = 1.20 Standard Deviation = 1.17 Relative Frequency of 1 Fault = 0.275 Total Frequency = 400</p> Signup and view all the answers

    Give an example of a scenario that can be represented by a continuous random variable.

    <p>The temperature in a room.</p> Signup and view all the answers

    How many total faults were recorded in the data?

    <p>400</p> Signup and view all the answers

    In a discrete probability distribution, each possible value of the random variable is associated with a probability referred to as P(____).

    <p>x</p> Signup and view all the answers

    The probability of getting 0 heads when tossing 2 coins is 0.50.

    <p>False</p> Signup and view all the answers

    Using the Python scipy.stats library, which function would you call to compute the variance?

    <p>discvar.var()</p> Signup and view all the answers

    For 3 faults, the probability P(x) is _____

    <p>0.225</p> Signup and view all the answers

    Which statement about a discrete random variable is true?

    <p>It can only take a finite number of values.</p> Signup and view all the answers

    A continuous random variable can assume only a finite number of values.

    <p>False</p> Signup and view all the answers

    What does the expected value E(x) represent in probability distributions?

    <p>The average or mean value of a discrete random variable.</p> Signup and view all the answers

    The ________ of a random variable measures the spread or dispersion in a set of data.

    <p>standard deviation</p> Signup and view all the answers

    Match the following examples with their corresponding type of random variable:

    <p>Number of customer complaints per day = Discrete Random Variable Temperature readings = Continuous Random Variable Number of defective items in a batch = Discrete Random Variable Height of students in a class = Continuous Random Variable</p> Signup and view all the answers

    What information is needed to calculate the expected value E(x) of a discrete random variable?

    <p>Values and their corresponding probabilities</p> Signup and view all the answers

    The standard deviation always has a higher value than the expected value.

    <p>False</p> Signup and view all the answers

    In the expected value formula, the variable x represents the ________ of the discrete random variable.

    <p>values</p> Signup and view all the answers

    Give one example of a discrete random variable.

    <p>Number of TVs in a household.</p> Signup and view all the answers

    What is the relative frequency of having 1 fault?

    <p>0.275</p> Signup and view all the answers

    The standard deviation of faults recorded is 1.17.

    <p>True</p> Signup and view all the answers

    What is the expected value E[x] for the given frequency distribution?

    <p>1.20</p> Signup and view all the answers

    The expected value is a measure of the _____ of a random variable.

    <p>central tendency</p> Signup and view all the answers

    Match the values of faults recorded with their respective probabilities:

    <p>0 faults = 0.375 1 fault = 0.275 2 faults = 0.125 3 faults = 0.225</p> Signup and view all the answers

    Which step is not involved in calculating the expected value of a discrete distribution?

    <p>Calculating the mean of the frequency</p> Signup and view all the answers

    The total probability of a discrete probability distribution must equal 1.

    <p>True</p> Signup and view all the answers

    What is the formula used to calculate the variance in discrete probability distributions?

    <p>Variance = Σ((x - E[x])² * P(x))</p> Signup and view all the answers

    In Python, the function used to compute the mean of a discrete variable is _____.

    <p>discvar.mean()</p> Signup and view all the answers

    Study Notes

    Introduction to Probability Distributions

    • A random variable is defined as a numerical value determined by the outcome of a random experiment, reflecting inherent randomness.

    Types of Random Variables

    • Discrete Random Variable: Takes on a finite or countably infinite set of values (e.g., 0, 1, 2,...).
    • Continuous Random Variable: Can assume values in an uncountable infinite spectrum.

    Examples of Discrete Random Variables

    • Many Possible Outcomes:
      • Number of daily complaints
      • Number of televisions in a household
      • Number of rings before a phone is answered
    • Only Two Possible Outcomes:
      • Defective item: Yes or No
      • Game result: Won or Lost

    Expected Value and Standard Deviation

    • Expected Value (E(x)):
      • Represents the average expected outcome for a discrete random variable.
      • Calculated as the sum of each value multiplied by its probability (E[x] = Σ[x * P(x)]).
    • Standard Deviation (σ):
      • Measures the dispersion or spread of values in a data set.
      • Indicates how much the values of a random variable deviate from the expected value.

    Example: ABC Ltd and Installation Faults

    • Frequency Distribution:

      • 0 faults: 150 occurrences
      • 1 fault: 110 occurrences
      • 2 faults: 50 occurrences
      • 3 faults: 90 occurrences
      • Total: 400 occurrences

    • Probability Distribution:

      • Calculated by dividing the frequency of each fault count by the total (e.g., 0 faults = 0.375).

    • Expected Value Computation:

      • E[x] calculated as 1.20, reflecting the average number of faults per installation.

    • Standard Deviation Computation:

      • σx calculated as √1.360 = 1.17, indicating the variability of the number of faults.

    Example: Tossing Two Coins

    • Probability Distribution:
      • 0 heads: P(x) = 0.25
      • 1 head: P(x) = 0.50
      • 2 heads: P(x) = 0.25
    • Expected Value: E(x) = 1.0.
    • Standard Deviation: σ = 0.707, illustrating the spread of outcomes.

    Python Example: Discrete Variables

    • To compute mean, variance, and standard deviation:
      • Use rv_discrete from scipy.stats.
      • Steps include defining the random variable values and their probabilities, linking them, and utilizing functions:
        • Compute mean: discvar.mean()
        • Compute variance: discvar.var()
        • Compute standard deviation: discvar.std()

    Introduction to Probability Distributions

    • A random variable is defined as a numerical value determined by the outcome of a random experiment, reflecting inherent randomness.

    Types of Random Variables

    • Discrete Random Variable: Takes on a finite or countably infinite set of values (e.g., 0, 1, 2,...).
    • Continuous Random Variable: Can assume values in an uncountable infinite spectrum.

    Examples of Discrete Random Variables

    • Many Possible Outcomes:
      • Number of daily complaints
      • Number of televisions in a household
      • Number of rings before a phone is answered
    • Only Two Possible Outcomes:
      • Defective item: Yes or No
      • Game result: Won or Lost

    Expected Value and Standard Deviation

    • Expected Value (E(x)):
      • Represents the average expected outcome for a discrete random variable.
      • Calculated as the sum of each value multiplied by its probability (E[x] = Σ[x * P(x)]).
    • Standard Deviation (σ):
      • Measures the dispersion or spread of values in a data set.
      • Indicates how much the values of a random variable deviate from the expected value.

    Example: ABC Ltd and Installation Faults

    • Frequency Distribution:

      • 0 faults: 150 occurrences
      • 1 fault: 110 occurrences
      • 2 faults: 50 occurrences
      • 3 faults: 90 occurrences
      • Total: 400 occurrences

    • Probability Distribution:

      • Calculated by dividing the frequency of each fault count by the total (e.g., 0 faults = 0.375).

    • Expected Value Computation:

      • E[x] calculated as 1.20, reflecting the average number of faults per installation.

    • Standard Deviation Computation:

      • σx calculated as √1.360 = 1.17, indicating the variability of the number of faults.

    Example: Tossing Two Coins

    • Probability Distribution:
      • 0 heads: P(x) = 0.25
      • 1 head: P(x) = 0.50
      • 2 heads: P(x) = 0.25
    • Expected Value: E(x) = 1.0.
    • Standard Deviation: σ = 0.707, illustrating the spread of outcomes.

    Python Example: Discrete Variables

    • To compute mean, variance, and standard deviation:
      • Use rv_discrete from scipy.stats.
      • Steps include defining the random variable values and their probabilities, linking them, and utilizing functions:
        • Compute mean: discvar.mean()
        • Compute variance: discvar.var()
        • Compute standard deviation: discvar.std()

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    Description

    This quiz covers the basics of probability distributions, including the definition and types of random variables, such as discrete and continuous random variables.

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