List of formulas related to continuous random variable and its functions:

  1. Continuous random variable ( X ):
  • This is a quantity that can take any value from a certain interval.
  1. Distribution Function (CDF) ( F(x) ):
  • Defined as:
    [
    F(x) = P(X \leq x)
    ]
  • This is the probability that a random variable ( X ) will take on a value less than or equal to ( x ).
  1. Probability Density (PDF) ( f(x) ):
  • It is related to the distribution function as follows:
    [
    f(x) = \frac{d}{dx} F(x)
    ]
  • This is the derivative of the distribution function, which shows what is the probability that a random variable will take a value in the neighborhood of ( x ).
  1. Properties of the distribution function:
  • ( F(x) ) is always in the range from 0 to 1:
    [
    0 \leq F(x) \leq 1
    ]
  • ( \lim_{x \to -\infty} F(x) = 0 )
  • ( \lim_{x \to +\infty} F(x) = 1 )
  1. Properties of probability density:
  • The probability density is always non-negative:
    [
    f(x) \geq 0
    ]
  • The integral of the probability density over the entire space is 1:
    [
    \int_{-\infty}^{+\infty} f(x) \, dx = 1
    ]

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