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Uniform Distribution quiz

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  • What is the main difference between discrete and continuous random variables?
    Discrete random variables can only take specific, separate values, while continuous random variables can take any value within a range.
  • How are probabilities represented for continuous random variables?
    Probabilities for continuous random variables are represented by the area under the probability density function over an interval.
  • Why can't we use a table to list all possible values of a continuous random variable?
    Because there are infinitely many possible values within any range, making it impossible to list them all in a table.
  • What is a probability density function (PDF)?
    A probability density function is a function that describes the likelihood of a continuous random variable taking on a range of values.
  • How do you find the probability that a continuous random variable falls within an interval?
    You calculate the area under the probability density function between the endpoints of the interval.
  • What is the probability that a continuous random variable equals a specific value?
    The probability is zero because there are infinitely many possible values, so the chance of hitting exactly one value is zero.
  • How is the total probability under a probability density function represented?
    The total area under the probability density function is always equal to one.
  • What is a uniform distribution?
    A uniform distribution is a continuous probability distribution where every value within a certain range has the same probability density.
  • How can you recognize a uniform distribution from its graph?
    The probability density function has the same height for every value within the range, forming a rectangle.
  • If a uniform distribution ranges from 0 to 6, what is the height of the probability density function?
    The height is 1 divided by the width of the range, so 1/6.
  • How do you calculate the probability that x is between 1 and 3 in a uniform distribution from 0 to 6?
    Multiply the height of the PDF (1/6) by the width of the interval (2), giving a probability of 2/6.
  • Why is the probability that x equals 5 zero in a continuous uniform distribution?
    Because the probability is based on area, and a single point has zero width, so its area and probability are zero.
  • What does it mean for a probability density function to have the same height everywhere?
    It means the distribution is uniform, with equal likelihood for all values in the range.
  • Can all continuous random variables be modeled with a uniform distribution?
    No, the uniform distribution is a special case; not all continuous random variables have equal probability density.
  • What is the sum of probabilities for all possible values of a continuous random variable?
    The sum, represented by the total area under the probability density function, is always one.