Below is my Google Docs Spreadsheet for Normal Distributions:
This sheet you cannot change because I have it locked, but if you would like access so you can use my sheet, and just put in your data and get the answer - email me: messenger1964@yahoo.com
| Normal Distribution | |||||
| a | μ | X | σ | Answer Φ(a) | |
| 4.4 | 2 | 1.6 | -1.5 | ||
| b | μ | X | σ | Answer Φ(b) | |
| 10 | 10.3 | 0.25 | 1.2 | ||
| Formula (p.d.f) | |||||
| f(x) = 1/(σ√2π)*e^(-(x-μ)^(2)/2σ^(2)) | a | ⇒ | ⇒ | ⇒ | 0.000000085818477 |
| ∀ x ∈ R (-∞, +∞) | |||||
| Google Sheets Formula = | (=(1/(2*PI())*EXP(-D3-C3)^(2))/2*E3^(2)) | ||||
| Converting P(X) to P(Z) (raw Z score) | Answer Z | ||||
| P((X-μ)/σ) | a | -1.5 | |||
| Example: P(X<10.1)= P((X-μ)/σ)<P(10.1-10/.25) | b | 1.2 | |||
| if u = 10, X = 10.1 and σ = .25 | |||||
| Using Standard Norm Table Formulas: | |||||
| P{Z<a} = Φ(a) | ⇒ | ⇒ | -1.5 | ||
| P{Z>a} = 1-Φ(a) | ⇒ | ⇒ | 2.5 | ||
| P{Z<-a} = 1-Φ(a) | ⇒ | ⇒ | 2.5 | ||
| P{Z>-a} = Φ(a) | ⇒ | ⇒ | -1.5 | ||
| P{a<Z<b} = Φ(b)-Φ(a) | a<b | ⇒ | ⇒ | 2.7 | |
| P{-a<Z<b} = Φ(b)+Φ(a)-1 | a<b | ⇒ | ⇒ | -1.3 | |
| P{-b<Z<a} = Φ(b)+Φ(a)-1 | a<b | ⇒ | ⇒ | -1.3 | |
| P{-b<Z<a} = Φ(b)-Φ(a) | a<b | ⇒ | ⇒ | 2.7 | |
| Finding Percentiles of Normal Distributions: | Z | μ | σ | Answer (X) | |
| Z = (X-μ)/σ so X = μ+σ*Z | -0.45 | 4.4 | 1.2 | ||
| X = μ+σ*Z = | |||||
| X= | 3.86 | ||||
| Google Sheets Formula = | (=E29+F29*D29) | ||||
| E[X] = | |||||
| E[X]=μ | ⇒ | ⇒ | ⇒ | 4.4 | |
| E[X^(2)] = | |||||
| σ^(2)+μ^(2) | ⇒ | ⇒ | ⇒ | 21.92 | |
| Google Sheets Formula = | (=E3^(2)+C3^(2)) | ||||
| Variance | |||||
| Var = σ^(2) = σ^(2) | ⇒ | ⇒ | ⇒ | 2.56 | |
| Standard Deviation | |||||
| St.Dev = √σ | ⇒ | ⇒ | ⇒ | 1.6 | |
| E[X^(3)] = | |||||
| 6*θ^(3) | ⇒ | ⇒ | ⇒ | 24.576 | |
| Third Moment of X about the Mean = | |||||
| E[(X-E[X])^(3)]= | |||||
| 2*θ^(3) | ⇒ | ⇒ | ⇒ | 8.192 | |
| Moment Generating Function | |||||
| M(t) = E[e^(tX)] = | |||||
| e^[μt+((σ^(2)*(t^(2))/2] | |||||
