Statistics Probability and Distribution Concepts
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Questions and Answers

What type of random variable is represented by the number of defective mobile phones in a box?

  • Neither discrete nor continuous
  • Discrete (correct)
  • Both discrete and continuous
  • Continuous

    • What is the symbol typically used to denote the mean of a probability distribution?

  • μ (correct)
  • ρ
  • σ²
  • σ

    • Which of the following satisfied the properties of a probability distribution?

  • P(x) = {0, 0.5, 0.5} (correct)
  • P(x) = {0.25, 0.25, 0.25, 0.25} (correct)
  • P(x) = {-0.2, 0.5, 0.4}
  • P(x) = {0.1, 0.2, 0.3, 0.5}

    • What are the possible outcomes of a binary distribution?

    <p>Success and failure</p> Signup and view all the answers

    In a binomial probability distribution with n = 25 and p = 0.8, which formula would you use to determine P(x = 15)?

    <p>P(x) = nCk * (p^k) * (1-p)^(n-k)</p> Signup and view all the answers

    What indicates that the random variable for the number of cars sold per day at a local dealership is discrete?

    <p>It is a countable value.</p> Signup and view all the answers

    What is NOT a characteristic of the variance in a probability distribution?

    <p>It can be negative.</p> Signup and view all the answers

    Which of the following is true about the number of flights in a day?

    <p>It is a discrete random variable.</p> Signup and view all the answers

    Which of the following is a continuous random variable?

    <p>The length of time in studying Statistics</p> Signup and view all the answers

    What theorem can be applied to find the variance of a binomial distribution?

    <p>Variance = np(1-p)</p> Signup and view all the answers

    Which of the following outcomes would invalidate the properties of a probability distribution?

    <p>P(x) = {0.2, 0.3, -0.5}</p> Signup and view all the answers

    If two dice are tossed, what is the probability of rolling a total of 12?

    <p>1/36</p> Signup and view all the answers

    What is the correct formula to calculate the mean of a discrete probability distribution?

    <p>Mean = Σ(x * P(x))</p> Signup and view all the answers

    In the context of discrete random variables, which of the following describes a valid sample space?

    <p>{0, 1, 2, 3, 4, 5}</p> Signup and view all the answers

    Which of the following statements correctly characterizes a binomial distribution?

    <p>It can only have two possible outcomes.</p> Signup and view all the answers

    Which of the following represents a proper binomial probability such that P(x) is valid?

    <p>P(x) = {0.25, 0.5, 0.25}</p> Signup and view all the answers

    Study Notes

    Random Variables

    • Discrete Variable: A variable that can be counted. Examples: Number of defective mobile phones in a box, the number of customers at McDonald's.
    • Continuous Variable: A variable that can take on any value within a range. Examples: Time spent studying Statistics and Probability, the length of a book, amount of water used to boil an egg.

    Probability Distribution

    • Properties of Probability Distribution:
      • Sum of all probabilities must equal 1.
      • Probability of each event must be between 0 and 1.
    • Mean (Expected Value) Symbol: μ
    • Variance Symbol: σ²
    • Standard Deviation Symbol: σ

    Formuåas

    • Mean: μ = Σ[x * P(x)]
    • Variance: σ² = Σ[(x - μ)² * P(x)]
    • Binomial Distribution: P(x) = (nCx) * p^x * q^(n-x), where:
      • n = number of trials
      • x = number of successes
      • p = probability of success
      • q = probability of failure
      • nCx = number of combinations of n things taken x at a time (n! / (x! * (n-x)!))

    Identifying Probability Distributions

    • Correct Probability Distributions:
      • A.P(x) = {1/3, 1/3, 1/3}
      • B.P(x) = {0.3, 0.5, 0.1, 0.1}
      • C.P(x) = {0.15, 0.25, 0.3, 0.25, 0.05}
    • Incorrect Probability Distribution:
      • D.P(x) = {-0.15, 0.25, 0.3, 0.30} (Because probability cannot be negative)

    Outcomes

    • Die Toss: The value when a three will appear is C. 3
    • Binary (Bernoulli) Distribution: The possible outcomes are D. success and failure.
    • Number of Defective Mobile Phones: The possible outcomes are D. {0, 1, 2, 3, 4, 5}

    Binomial Distribution Calculation

    • n = 25, p = 0.8, find P(x = 15):
      • Use the formula to calculate the probability of exactly 15 successes in 25 trials.

    Calculating Mean, Variance, and Standard Deviation

    • Car Dealership Problem:
      • Construct a table with columns for x (number of cars sold), P(x), x * P(x), (x - μ)² * P(x).
      • Calculate the mean (μ) using the formula: μ = Σ[x * P(x)].
      • Calculate the variance (σ²) using the formula: σ² = Σ[(x - μ)² * P(x)].
      • Calculate the standard deviation (σ): σ = √σ².

    Dice Toss

    • Two dice tossed: The probability of getting any combination can be calculated by making a table listing all the possible outcomes. (See the table for detailed solution).

    Random Variables

    • Discrete: This type of variable can only take on a finite number of values (whole numbers or integers).
    • Continuous: This type of variable can take on an infinite number of values within a given range.
    • Example:
      • The number of defective mobile phones in a box is a discrete variable.
      • The number of customers at a McDonald's branch is a discrete variable.
      • The time spent studying is a continuous variable.
      • The number of reference books used is a discrete variable.
      • The amount of water used to boil an egg is a continuous variable.
      • The number of flights in a day is a discrete variable.

    Probability Distribution

    • Two properties:
      • The sum of all probabilities must be equal to 1.
      • Each probability must be between 0 and 1.

    Symbols

    • Mean: µ
    • Variance: σ²
    • Standard Deviation: σ

    Formulas

    • Mean (μ): μ = Σ(x * P(x))
    • Variance (σ²): σ² = Σ[(x - μ)² * P(x)]
    • Binomial Distribution: P(x) = (nCx) * p^x * q^(n-x)

    Binomial Distributions

    • Outcomes: Success and failure.
    • Probability of success: p
    • Probability of failure: q = 1 - p

    Example: Defective Phones

    • The possible number of defective phones in a box of five is: {0, 1, 2, 3, 4, 5}
    • The binomial distribution with n=25 and p=0.8 represents the probability of getting a certain number of successes (x) in 25 trials where the probability of success is 0.8.

    Car Dealership Example

    • The mean is calculated by summing the product of each number of cars sold and its corresponding probability.
    • The variance is calculated by summing the product of the squared difference between each number of cars sold and the mean, and its corresponding probability.
    • The standard deviation is the square root of the variance.

    Two Dice Example

    • The possible outcomes of rolling two dice can be represented using a table, showing the sum of the values on each die.

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    Description

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