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SOME IMPORTANT CONTINUOUS RANDOM VARIABLES Małgorzata Murat UNIFORM DISTRIBUTION Random variable X has uniform distribution U(a, b) if its probability density function is given by 1 , x ∈ (a, b) f (x) = b − a 0, x ∈ R r (a, b) Uniform distribution is very simple and used to model errors in electrical communication with pulse code modulation. 1 expected mean: EX = (a + b) 2 1 variance: σ 2 = (b − a)2 12 EXPONENTIAL DISTRIBUTION Random variable X has exponential distribution with parameter λ > 0 if its probability density function is given by f (x) = λe−λx , 0, x0 x<0 EXPONENTIAL DISTRIBUTION The exponential distribution is used for studies of reliability and of queuing theory, which gives probability as a function of waiting time in a queue for service is related to the Poisson distribution gives the probability distribution of the time between successive random events for the same conditions as apply to the Poisson distribution. EXPONENTIAL DISTRIBUTION expected mean: EX = λ variance: σ 2 = λ2 NORMAL DISTRIBUTION Random variable X has normal distribution with parameters µ, σ > 0 if its probability density function is given by f (x) = σ 1 √ 2π (x − µ)2 − 2σ 2 e expected mean: EX = µ determines the location of the center of the distribution variance: σ 2 determines the spread of the distribution