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Questions and Answers
What type of random variable is represented by the number of defective mobile phones in a box?
What is the symbol typically used to denote the mean of a probability distribution?
Which of the following satisfied the properties of a probability distribution?
What are the possible outcomes of a binary distribution?
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In a binomial probability distribution with n = 25 and p = 0.8, which formula would you use to determine P(x = 15)?
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What indicates that the random variable for the number of cars sold per day at a local dealership is discrete?
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What is NOT a characteristic of the variance in a probability distribution?
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Which of the following is true about the number of flights in a day?
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Which of the following is a continuous random variable?
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What theorem can be applied to find the variance of a binomial distribution?
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Which of the following outcomes would invalidate the properties of a probability distribution?
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If two dice are tossed, what is the probability of rolling a total of 12?
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What is the correct formula to calculate the mean of a discrete probability distribution?
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In the context of discrete random variables, which of the following describes a valid sample space?
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Which of the following statements correctly characterizes a binomial distribution?
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Which of the following represents a proper binomial probability such that P(x) is valid?
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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
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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)]
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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
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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}
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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
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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
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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.
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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
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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
Test your understanding of random variables, probability distributions, and essential statistical formulas. This quiz covers discrete and continuous variables, as well as properties of probability distributions and related calculations. Perfect for students wanting to solidify their grasp on statistics concepts.
