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Uniform Distribution

Uniform Distribution

Mar 18, 2024

Uniform Distribution

  • is used in any situation when a value is picked “at random” from a given interval; that is, without any preference for lower, higher, or medium values

Probability Density Function

Expectation

𝐄[𝑋] = (𝑏 + 𝑎) / 2

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  • 𝐄[𝑋] = -∞∫∞ 𝑥·𝑃𝐷𝐹(𝑋=𝑥)·𝑑𝑥
  • 𝐄[𝑋] = 𝑎∫𝑏 𝑥·[1/(𝑏 - 𝑎)]·𝑑𝑥
  • 𝐄[𝑋] = [1/(𝑏 - 𝑎)] 𝑎∫𝑏 𝑥·𝑑𝑥
  • 𝐄[𝑋] = [1/(𝑏 - 𝑎)]·(1/2)·[𝑥2]𝑎𝑏
  • 𝐄[𝑋] = [1/(𝑏 - 𝑎)]·(1/2)·(𝑏2-𝑎2)
  • 𝐄[𝑋] = [1/(𝑏 - 𝑎)]·(1/2)·(𝑏-𝑎)·(𝑏+𝑎)
  • 𝐄[𝑋] = (1/2)·(𝑏+𝑎)
  • 𝐄[𝑋] = (𝑏+𝑎)/2

Variance

𝑉𝑎𝑟(𝑋) = (𝑏 - 𝑎)2 / 12

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  • 𝑉𝑎𝑟(𝑋) = 𝐄[(𝑋 - 𝐄[𝑋])2] # as defined in Second Central Moment
  • 𝑉𝑎𝑟(𝑋) = -∞∫∞ (𝑥 - 𝐄[𝑋])2·𝑃𝐷𝐹(𝑋=𝑥)·𝑑𝑥
  • 𝑉𝑎𝑟(𝑋) = -∞∫∞ [𝑥 - ((𝑏+𝑎)/2)]2·𝑃𝐷𝐹(𝑋=𝑥)·𝑑𝑥
  • 𝑉𝑎𝑟(𝑋) = 𝑎∫𝑏 [𝑥 - ((𝑏+𝑎)/2)]2·[1/(𝑏 - 𝑎)]·𝑑𝑥
  • 𝑉𝑎𝑟(𝑋) = [1/(𝑏 - 𝑎)] 𝑎∫𝑏 [𝑥 - ((𝑏+𝑎)/2)]2·𝑑𝑥
  • 𝑉𝑎𝑟(𝑋) = [1/(𝑏 - 𝑎)] 𝑎∫𝑏 [𝑥2 - 𝑥(𝑏+𝑎) + ((𝑏+𝑎)/2)2]·𝑑𝑥
  • 𝑉𝑎𝑟(𝑋) = [1/(𝑏 - 𝑎)] 𝑎∫𝑏 [𝑥2 - 𝑥(𝑏+𝑎) + ((𝑏+𝑎)2/4)]·𝑑𝑥
  • 𝑉𝑎𝑟(𝑋) = [1/(𝑏 - 𝑎)] · [(1/3)𝑥3 - (1/2)𝑥2(𝑏+𝑎) + 𝑥((𝑏+𝑎)2/4)]𝑎𝑏
  • 𝑉𝑎𝑟(𝑋) = [1/(𝑏 - 𝑎)] · [[(1/3)𝑏3 - (1/2)𝑏2(𝑏+𝑎) + 𝑏((𝑏+𝑎)2/4)] - [(1/3)𝑎3 - (1/2)𝑎2(𝑏+𝑎) + 𝑎((𝑏+𝑎)2/4)]]
  • 𝑉𝑎𝑟(𝑋) = [1/(𝑏 - 𝑎)] · [[(𝑏3/3) - (𝑏3+𝑎𝑏2)/2 + (𝑏3+2𝑎𝑏2+𝑏𝑎2)/4] - [(𝑎3/3) - (𝑏𝑎2+𝑎3)/2 + (𝑎𝑏2+2𝑎2𝑏+𝑎3)/4)]]
  • 𝑉𝑎𝑟(𝑋) = [1/(𝑏 - 𝑎)] · [[(4𝑏3/12) - (6𝑏3+6𝑎𝑏2)/12 + (3𝑏3+6𝑎𝑏2+3𝑏𝑎2)/12] - [(4𝑎3/12) - (6𝑏𝑎2+6𝑎3)/12 + (3𝑎𝑏2+6𝑎2𝑏+3𝑎3)/12)]]
  • 𝑉𝑎𝑟(𝑋) = [1/(𝑏 - 𝑎)] · [1/12] · [[4𝑏3 - 6𝑏3 - 6𝑎𝑏2 + 3𝑏3 + 6𝑎𝑏2 + 3𝑏𝑎2] - [4𝑎3 - 6𝑏𝑎2 - 6𝑎3 + 3𝑎𝑏2 + 6𝑎2𝑏 + 3𝑎3]]
  • 𝑉𝑎𝑟(𝑋) = [1/12(𝑏 - 𝑎)] · [4𝑏3 - 6𝑏3 - 6𝑎𝑏2 + 3𝑏3 + 6𝑎𝑏2 + 3𝑏𝑎2 - 4𝑎3 + 6𝑏𝑎2 + 6𝑎3 - 3𝑎𝑏2 - 6𝑎2𝑏 - 3𝑎3]
  • 𝑉𝑎𝑟(𝑋) = [1/12(𝑏 - 𝑎)] · [𝑏3+ 3𝑏𝑎2 - 𝑎3 - 3𝑎𝑏2]
  • 𝑉𝑎𝑟(𝑋) = [1/12(𝑏 - 𝑎)] · [𝑏3 + 3𝑏𝑎2 - 3𝑎𝑏2 - 𝑎3]
    • [𝑏 - 𝑎]3
    • [𝑏 - 𝑎][𝑏 - 𝑎][𝑏 - 𝑎]
    • [𝑏2 - 2𝑎𝑏 + 𝑎2][𝑏 - 𝑎]
    • 𝑏[𝑏2 - 2𝑎𝑏 + 𝑎2] - 𝑎[𝑏2 - 2𝑎𝑏 + 𝑎2]
    • [𝑏3 - 2𝑎𝑏2 + 𝑏𝑎2] - [𝑎𝑏2 - 2𝑎2𝑏 + 𝑎3]
    • 𝑏3 - 2𝑎𝑏2 + 𝑏𝑎2 - 𝑎𝑏2 + 2𝑎2𝑏 - 𝑎3
    • 𝑏3 - 3𝑎𝑏2 + 3𝑏𝑎2 - 𝑎3
  • 𝑉𝑎𝑟(𝑋) = [1/12(𝑏 - 𝑎)] · [𝑏 - 𝑎]3
  • 𝑉𝑎𝑟(𝑋) = (𝑏 - 𝑎)2/12