Probability Distributions and PMF
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Questions and Answers

What is a probability distribution of a random variable?

The relationship between the values that the random variable takes and the probabilities of taking those values.

What is the probability mass function?

A table, formula, or graphical representation of the probabilities for each possible value of a discrete random variable.

What are the two conditions that the probability mass function must satisfy?

  • The sum of all probabilities must equal 1. (correct)
  • The probability of each value must be between 0 and 1. (correct)
  • The probability of any value must be greater than 0.
  • The sum of all probabilities must be less than 1.

    • If X represents the number of heads when three coins are thrown, what are the possible values of X?

    <p>The possible values of X are 0, 1, 2, and 3.</p> Signup and view all the answers

    What is the probability of getting at least one head when three coins are thrown?

    <p>7/8 (C)</p> Signup and view all the answers

    What is the probability of getting between 1 and 3 (excluding 3) heads when three coins are thrown?

    <p>6/8 (C)</p> Signup and view all the answers

    What is the probability of getting less than 2 heads when three coins are thrown?

    <p>4/8 (B)</p> Signup and view all the answers

    What is the mean (expected value) of a probability mass function?

    <p>The sum of each value multiplied by its corresponding probability.</p> Signup and view all the answers

    What is the variance of a probability mass function?

    <p>The expected value of the squared deviation of the random variable from its mean.</p> Signup and view all the answers

    What is the standard deviation of a probability mass function?

    <p>The square root of the variance.</p> Signup and view all the answers

    What is the probability density function?

    <p>A formula or graphical representation of the probabilities for each value of a continuous random variable.</p> Signup and view all the answers

    What are the two conditions that the probability density function must satisfy?

    <p>The function must be non-negative for all values. (A), The integral of the function over its entire domain must equal 1. (C)</p> Signup and view all the answers

    What is the mean (expected value) of a probability density function?

    <p>The integral of x multiplied by the probability density function over its entire domain.</p> Signup and view all the answers

    What is the variance of a probability density function?

    <p>The integral of (x - μ)² multiplied by the probability density function over its entire domain.</p> Signup and view all the answers

    What is the standard deviation of a probability density function?

    <p>The square root of the variance.</p> Signup and view all the answers

    Which of these are special probability distributions?

    <p>All of the above (D)</p> Signup and view all the answers

    What are the conditions for a binomial experiment?

    <p>The experiment must be repeated a fixed number of times (trials), each trial must have only two outcomes (success or failure), the probability of success must be constant for each trial, and the trials must be independent.</p> Signup and view all the answers

    What is the probability mass function for a binomial distribution?

    <p>P(X = x) = C(n, x) * p^x * q^(n-x), where n is the number of trials, x is the number of successes, p is the probability of success on a single trial, and q is the probability of failure on a single trial.</p> Signup and view all the answers

    What are the mean and variance of a binomial distribution?

    <p>The mean is E(X) = n * p, and the variance is V(X) = n * p * q.</p> Signup and view all the answers

    What does "at least a" mean in the context of probability?

    <p>The probability of getting a or more successes (D)</p> Signup and view all the answers

    What does "more than a" mean in the context of probability?

    <p>The probability of getting more than a successes (A)</p> Signup and view all the answers

    What does "between a and b" mean in the context of probability?

    <p>The probability of getting x successes where x is between a and b (excluding a and b) (A)</p> Signup and view all the answers

    What does "between a and b, inclusive" mean in the context of probability?

    <p>The probability of getting x successes where x is between a and b (including a and b) (A)</p> Signup and view all the answers

    What is the complement rule of probability?

    <p>The probability of an event occurring is equal to 1 minus the probability of the event not occurring.</p> Signup and view all the answers

    What is the Poisson distribution?

    <p>A discrete probability distribution that models the probability of a certain number of events occurring in a fixed interval of time or space, given that these events occur independently at a constant average rate.</p> Signup and view all the answers

    What is the probability mass function for a Poisson distribution?

    <p>P(X = x) = (e^(-λ) * λ^x) / x!, where λ is the average rate of events, and x is the number of events.</p> Signup and view all the answers

    What are the mean and variance of a Poisson distribution?

    <p>Both the mean and variance of a Poisson distribution are equal to λ.</p> Signup and view all the answers

    What is the normal (Gaussian) distribution?

    <p>A continuous probability distribution that is bell-shaped and symmetric, and widely used to model real-world phenomena like height, weight, or blood pressure.</p> Signup and view all the answers

    What are the parameters of a normal distribution?

    <p>The mean (μ) and the variance (σ²), which determine the location and spread of the distribution, respectively.</p> Signup and view all the answers

    What are the properties of the normal distribution?

    <p>The normal curve is bell-shaped and symmetrical, the mean, median, and mode of the distribution are all equal, it is completely defined by its mean and variance, and its location shifts with the mean while its spread changes with the variance.</p> Signup and view all the answers

    What is the standard normal distribution?

    <p>A special case of the normal distribution with a mean of 0 and a variance of 1.</p> Signup and view all the answers

    What is the Z-score?

    <p>A standardized score that measures how many standard deviations a data point is away from the mean of the standard normal distribution.</p> Signup and view all the answers

    How can we calculate the probability of events using the standard normal distribution?

    <p>By using the Z-score and looking up the probability in a standard normal table (Z-table) or by using statistical software or calculators.</p> Signup and view all the answers

    Flashcards

    Probability distribution

    A probability distribution describes how likely each possible value of a random variable is to occur.

    Probability mass function (PMF)

    A probability mass function (PMF) assigns a probability to each possible value of a discrete random variable.

    Discrete random variable

    A discrete random variable can only take on a finite number of values or a countably infinite number of values.

    Probability range for PMF

    The probability of a specific value in a probability mass function must always be between 0 and 1.

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    Sum of probabilities in PMF

    The sum of all probabilities in a probability mass function must equal 1.

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    Representations of PMF

    A probability mass function can be represented as a table, formula, or a graph.

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    Notation of PMF

    The probability of a specific value in a probability mass function is represented by P(x), where x is the specific value.

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    Understanding outcomes with probability distribution

    A probability distribution can help you understand the possible outcomes of a random event and their likelihood.

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    Random variable

    A random variable can be described as a variable whose value is a numerical outcome of a random phenomenon.

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    Applications of probability distributions

    Probability distributions are used in various fields, including statistics, finance, and machine learning, to understand and analyze data.

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    Study Notes

    Probability Distributions

    • A probability distribution is a relationship between the values a random variable takes and their corresponding probabilities.
    • For discrete random variables, this can be expressed as a table, formula, or graphically.
    • The probability mass function (PMF) must satisfy two conditions:
      • 0 ≤ P(xᵢ) ≤ 1
      • ΣP(xᵢ) = 1

    Probability Mass Function (PMF) Example

    • Scenario: Find the probability mass function (PMF) when three coins are tossed and 'x' represents the number of heads.
    • Possible Outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT
    • Possible Values of X: 0, 1, 2, 3
    • Probabilities:
      • P(X=0) = 1/8
      • P(X=1) = 3/8
      • P(X=2) = 3/8
      • P(X=3) = 1/8

    Additional PMF Examples

    • Finding probabilities: Given a PMF table, calculate specific probabilities like P(x ≥ 1), P(1 ≤ x < 3), and P(x < 2).

    Expected Value (Mean)

    • The mean or expected value (E(x)) of a discrete random variable is calculated by summing the product of each value xᵢ and its probability P(xᵢ).
      • E(X) = Σ [ xᵢ * P(xᵢ)]

    Variance

    • Variance (V(x)) measures the spread or dispersion of a probability distribution. It's calculated by finding the expected value of the squared difference between each value and the mean
    • V(x) = E[ (x - E(x))² ] = Σ [(xᵢ - E(x))² * P(xᵢ)]

    Standard Deviation

    • The standard deviation is the square root of the variance, and provides another measure of the spread
    • SD(x) = √V(x)

    Probability Density Function (PDF)

    • For continuous random variables, probability is represented using a probability density function (PDF). It must satisfy two conditions:
      • f(x) ≥ 0
      • ∫ f(x) dx = 1

    Finding Probabilities Using PDF

    • To find probabilities for a continuous random variable, calculate the definite integral of the PDF over the desired range.
    • Calculate mean and variance of a probability density function (PDF). Use formulas involving integrals.

    Binomial Distribution

    • A binomial distribution describes the probability of exactly 'k' successes in 'n' independent trials, given a fixed probability of success 'p' for each trial.
    • P(x=k) = (n choose k) * p^k * (1-p)^(n-k)
    • Mean = np
    • Variance = np(1-p)

    Poisson Distribution

    • The Poisson distribution describes the probability of a given number of events occurring in a fixed interval of time or space, given the average rate at which the events happen.
    • P(x=k) = (λ^k * e^-λ) / k!
    • Mean = λ
    • Variance = λ

    Normal Distribution

    • The Gaussian or normal distribution is a continuous probability distribution, often shaped like a bell curve.
    • It is defined by the mean (μ) and the variance (σ²).
    • Many real-world phenomena follow a normal distribution.
    • The z-score converts any normal distribution to a standard normal distribution (mean=0, standard deviation =1).

    Calculating Probabilities for Normal Distributions

    • Use the z-table or calculator to calculate probabilities given a mean and standard deviation.

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    Related Documents

    CH 7 - Statistics PDF

    Description

    This quiz explores key concepts of probability distributions, focusing on the probability mass function (PMF) for discrete random variables. You'll learn how to identify possible outcomes and calculate probabilities for different scenarios, including practical examples like tossing coins. Test your understanding of expected value and PMF calculations.

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