Uniform Distribution
- is used in any situation when a value is picked βat randomβ from a given interval; that is, without any preference for lower, higher, or medium values
Probability Density Function
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Expectation
π[π]Β = (π + π) / 2
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- π[π]Β = -ββ«β π₯Β·ππ·πΉ(π=π₯)Β·ππ₯
- π[π]Β = πβ«π π₯Β·[1/(π - π)]Β·ππ₯
- π[π]Β = [1/(π - π)] πβ«π π₯Β·ππ₯
- π[π]Β = [1/(π - π)]Β·(1/2)Β·[π₯2]ππ
- π[π]Β = [1/(π - π)]Β·(1/2)Β·(π2-π2)
- π[π]Β = [1/(π - π)]Β·(1/2)Β·(π-π)Β·(π+π)
- π[π]Β = (1/2)Β·(π+π)
- π[π]Β = (π+π)/2
Variance
πππ(π) = (π - π)2 / 12
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- πππ(π) = π[(π - π[π])2] # as defined in Second Central Moment
- πππ(π) = -ββ«β (π₯ - π[π])2Β·ππ·πΉ(π=π₯)Β·ππ₯
- πππ(π) = -ββ«β [π₯ - ((π+π)/2)]2Β·ππ·πΉ(π=π₯)Β·ππ₯
- πππ(π) = πβ«π [π₯ - ((π+π)/2)]2Β·[1/(π - π)]Β·ππ₯
- πππ(π) = [1/(π - π)] πβ«π [π₯ - ((π+π)/2)]2Β·ππ₯
- πππ(π) = [1/(π - π)] πβ«π [π₯2 - π₯(π+π) + ((π+π)/2)2]Β·ππ₯
- πππ(π) = [1/(π - π)] πβ«π [π₯2 - π₯(π+π) + ((π+π)2/4)]Β·ππ₯
- πππ(π) = [1/(π - π)] Β· [(1/3)π₯3 - (1/2)π₯2(π+π) + π₯((π+π)2/4)]ππ
- πππ(π) = [1/(π - π)] Β· [[(1/3)π3 - (1/2)π2(π+π) + π((π+π)2/4)] - [(1/3)π3 - (1/2)π2(π+π) + π((π+π)2/4)]]
- πππ(π) = [1/(π - π)] Β· [[(π3/3) - (π3+ππ2)/2 + (π3+2ππ2+ππ2)/4] - [(π3/3) - (ππ2+π3)/2 + (ππ2+2π2π+π3)/4)]]
- πππ(π) = [1/(π - π)] Β· [[(4π3/12) - (6π3+6ππ2)/12 + (3π3+6ππ2+3ππ2)/12] - [(4π3/12) - (6ππ2+6π3)/12 + (3ππ2+6π2π+3π3)/12)]]
- πππ(π) = [1/(π - π)] Β· [1/12] Β· [[4π3 - 6π3- 6ππ2 + 3π3+ 6ππ2+ 3ππ2] - [4π3 - 6ππ2- 6π3 + 3ππ2+ 6π2π + 3π3]]
- πππ(π) = [1/12(π - π)] Β· [4π3 - 6π3- 6ππ2 + 3π3+ 6ππ2+ 3ππ2 - 4π3 + 6ππ2 + 6π3 - 3ππ2- 6π2π - 3π3]
- πππ(π) = [1/12(π - π)] Β· [π3+ 3ππ2 - π3 - 3ππ2]
- πππ(π) = [1/12(π - π)] Β· [π3+ 3ππ2 - 3ππ2 - π3]
- [π - π]3
- [π - π][π - π][π - π]
- [π2 - 2ππ + π2][π - π]
- π[π2 - 2ππ + π2] - π[π2 - 2ππ + π2]
- [π3 - 2ππ2 + ππ2] - [ππ2 - 2π2π + π3]
- π3 - 2ππ2 + ππ2 - ππ2 + 2π2π - π3
- π3 - 3ππ2 + 3ππ2 - π3
- πππ(π) = [1/12(π - π)] Β· [π - π]3
- πππ(π) = (π - π)2/12