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Statistics -Continuous probability distribution. 201 3 /11/18. Probability density function. With continuous ransom variables, the counterpart of the probability function is the probability density function, denoted by f ( x ) <Note> How to compute Pr ( a≤x≤b )?

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probability density function
Probability density function
  • With continuous ransom variables, the counterpart of the probability function is the probability density function, denoted by f(x)

<Note> How to compute Pr(a≤x≤b)?

<Note>The probability of any particular value of the continuous random variable is zero.

continuous probability distribution
Continuous probability distribution
  • For a continuous random variable x:
    • The probability distribution is defined by a probability density function, denoted by

f(x)

    • The expected value of a continuous random variable is a measure of the central location for the random variable.
    • The variance is used to summarize the variability in the values of a random variable.
uniform probability distribution
Uniform probability distribution
  • Uniform probability density function:
  • Expected value for uniform probability distribution:
  • Variance for uniform probability distribution:

f (x) = 1/(b – a) for a<x<b

= 0 elsewhere

E(x) = (a + b)/2

Var(x) = (b - a)2/12

normal probability distribution
Normal probability distribution
  • Normal probability density function:
  • Expected value for normal probability distribution:
  • Variance for normal probability distribution:
standard normal probability distribution
Standard normal probability distribution
  • Standard normal probability density function:
  • Expected value for standard normal probability distribution:

0

  • Variance for standard normal probability distribution:

1

exponential probability distribution
Exponential probability distribution
  • Exponential probability density function:
  • Expected value for exponential probability distribution:
  • Variance for exponential probability distribution:
other distributions
Other distributions
  • Chi-square distribution
  • t distribution
  • F distribution
  • others
relationships between distributions
Relationships between distributions
  • Normal distribution vs. Standard normal distribution
  • Normal distribution vs. Binomial distribution
  • Poisson distribution vs. Exponential distribution