Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
SOME SIMPLE PROBABILITY DENSITY FUNCTIONS (PDF) f ( x) = UNIFORM 1 β −α α ≤x≤β The uniform distribution is useful in representing random variables which have known upper and lower bounds and which have equal likelihood of occurring anywhere between these bounds. That is, if you know nothing else about the relative likelihood of a random variable, aside from its upper and lower bounds, then the uniform distribution is appropriate -- it makes no assumptions regarding preferential likelihood of the random variable since all possible values are equilikely. f ( x) E[ X ] = α +β 2 0.25 ( β − a)2 Var[ X ] = 12 0 1 2 3 4 5 6 7 8 9 x EXPONENTIAL E[T ] = ⎧λ e − λt , for t ≥ 0 fT (t ) = ⎨ ⎩ 0, otherwise 1 λ Var[T ] = 1 λ2 Suppose that the time-to-failure, T, of a clay barrier has the probability density function; ⎧λ e − λt , for t ≥ 0 fT (t ) = ⎨ ⎩ 0, otherwise 0.015 0.020 where λ = 0.02/year, then what is the probability that T exceeds 100 years? ∞ 0.010 ∫ 100 fT (t ) dt = ∫ λe 100 0.005 = e −100 λ = e −100(0.02) = e −2 = 0.1353 P[ T > 100 ] 0 fT (t) P[ T > 100 ] = ∞ 0 50 100 t (years) 150 200 − λt dt NORMAL The best known Probability Density Function is the Normal or Gaussian distribution. Let X be a normally distributed random variable with mean and standard deviation given by µ X and σ X . In this case the PDF is given by: 2 ⎧ ⎛ ⎞ 1 ⎪ 1 x − µ X ⎫⎪ exp ⎨− ⎜ f X ( x) = ⎟ ⎬ 2 σ σ X 2π X ⎠ ⎭⎪ ⎪⎩ ⎝ E[ X ] = µ X Var[ X ] = σ X2 NORMAL DISTRIBUTION µ X = 100 σ X = 50 f X ( x) Mean,median and mode The area under the distribution is unity inflection points are at µ ± σ x