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Statistics

# Formula Review

StatisticsFormula Review

### 5.1Continuous Probability Functions

Probability density function (pdf) f(x):

• f(x) ≥ 0
• The total area under the curve f(x) is one.

Cumulative distribution function (cdf): P(Xx)

### 5.2The Uniform Distribution

X = a real number between a and b (in some instances, X can take on the values a and b). a = smallest X, b = largest X

X ~ U(a, b)

The mean is $μ= a+b 2 . μ= a+b 2 .$

The standard deviation is

Probability density function: $f(x)= 1 b−a f(x)= 1 b−a$ for $a≤X≤b a≤X≤b$

Area to the left of x: P(X < x) = (xa)$( 1 b−a ) ( 1 b−a )$

Area to the right of x: P(X > x) = (bx)$( 1 b−a ) ( 1 b−a )$

Area between c and d: P(c < x < d) = (base)(height) = (dc)$( 1 b−a ) ( 1 b−a )$

Uniform: X ~ U(a, b) where a < x < b

• pdf: $f( x )= 1 b−a f( x )= 1 b−a$ for a ≤ x ≤ b
• cdf: P(Xx) = $x−a b−a x−a b−a$
• mean µ = $a+b 2 a+b 2$
• standard deviation σ $= (b−a) 2 12 = (b−a) 2 12$
• P(c < X < d) = (dc)$( 1 b–a ) ( 1 b–a )$

### 5.3The Exponential Distribution (Optional)

Exponential: X ~ Exp(m) where m = the decay parameter

• pdf: f(x) = me(–mx) where x ≥ 0 and m > 0
• cdf: P(Xx) = 1 – e(–mx)
• mean µ = $1 m 1 m$
• standard deviation σ = µ
• percentile k: k = $ln(1−AreaToTheLeftOfk) (−m) ln(1−AreaToTheLeftOfk) (−m)$
• Additionally
• P(X > x) = e(–mx)
• P(a < X < b) = e(–ma)e(–mb)
• Memoryless property: P(X > x + k|X > x) = P (X > k)
• Poisson probability: with mean λ
• k! = k*(k−1)*(k−2)*(k−3)*…3*2*1
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