(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 (B)

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 (B)</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 (C)</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 (A)</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 (B)</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 (B)</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. (C)</p> Signup and view all the answers

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

<p>False (B)</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 (D)</p> Signup and view all the answers

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

<p>False (B)</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 (D)</p> Signup and view all the answers

The standard deviation of faults recorded is 1.17.

<p>True (A)</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 (A)</p> Signup and view all the answers

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

<p>True (A)</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

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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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