Chapter 5: Continuous Random Variables

**Probability density function**: a statistical measure used to gauge the likely outcome of a discrete value*f*(*x*) ≥ 0The total area under the curve

*f*(*x*) is one.

**Cumulative distribution****function**: a function whose value is the probability that a corresponding continuous random variable has a value less than or equal to the argument of the function.*P*(*X*≤*x*) = (𝑥−𝑎)/(𝑏−𝑎)

**Continuous probability distributions:**PROBABILITY = AREA**Continuous probability density function**: gives the relative likelihood of any outcome in a continuum occurring𝑓(𝑥)=1 / 𝑏−𝑎 for

*a*≤*x*≤*b*

**The****uniform distribution**: is a continuous probability distribution and is concerned with events that are equally likely to occur.**Uniform Mean**: 𝜇=(𝑎+𝑏) / 2**Uniform Standard deviation**: 𝜎=√((𝑏−𝑎)^2 / 12)**Uniform pdf**: 𝑓(𝑥)=1 / 𝑏−𝑎 for*a ≤ x ≤ b***Uniform cdf**:*P*(*X*≤*x*) = 𝑥−𝑎 / 𝑏−𝑎

*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***Probability density function:**𝑓(𝑥)=(1 / b−a) for 𝑎 ≤ 𝑋 ≤ 𝑏**Area to the Left of****:***x**P*(*X*<*x*) = (*x*–*a*)(1 / 𝑏−𝑎)**Area to the Right of****:***x**P*(*X*>*x*) = (*b*–*x*)(1 / 𝑏−𝑎)**Area Between***c***and****:***d**P*(*c*<*x*<*d*) = (base)(height) = (*d*–*c*)(1 / 𝑏−𝑎)

**Memoryless property**: the independence of events or, more specifically, the independence of event-to-event times or*P*(*X*>*r*+*t*|*X*>*r*) =*P*(*X*>*t*) for all*r*≥ 0 and*t*≥ 0**Exponential Distribution**:*X*~*Exp*(*m*) where*m*= the decay parameter**decay parameter**:*m*= 1 / μ and we write*X*∼*Exp*(*m*) where*x*≥ 0 and*m*> 0**exponential pdf**:*f*(*x*) =*me^*(–*mx*) where*x*≥ 0 and*m*> 0**exponential cdf**:*P*(*X*≤*x*) = 1 –*e^*(–*mx*)**exponential mean**:*µ*= 1/𝑚**exponential standard deviation**:*σ*=*µ***exponential percentile***k*:*k*= 𝑙𝑛(1−𝐴𝑟𝑒𝑎𝑇𝑜𝑇ℎ𝑒𝐿𝑒𝑓𝑡𝑂𝑓𝑘) / (−𝑚)Additionally

*P*(*X*>*x*) =*e^*(–*mx*)*P*(*a*<*X*<*b*) =*e^*(–*ma*) –*e^*(–*mb*)

**Poisson probability**: 𝑃(𝑋=𝑘)=𝜆^𝑘 𝑒^−𝑘 / 𝑘! P(X=k) with mean*λ***!** =*k**k**(*k*-1)*(*k*-2)*(*k*-3)*…3*2*1

**Probability density function**: a statistical measure used to gauge the likely outcome of a discrete value*f*(*x*) ≥ 0The total area under the curve

*f*(*x*) is one.

**Cumulative distribution****function**: a function whose value is the probability that a corresponding continuous random variable has a value less than or equal to the argument of the function.*P*(*X*≤*x*) = (𝑥−𝑎)/(𝑏−𝑎)

**Continuous probability distributions:**PROBABILITY = AREA**Continuous probability density function**: gives the relative likelihood of any outcome in a continuum occurring𝑓(𝑥)=1 / 𝑏−𝑎 for

*a*≤*x*≤*b*

**The****uniform distribution**: is a continuous probability distribution and is concerned with events that are equally likely to occur.**Uniform Mean**: 𝜇=(𝑎+𝑏) / 2**Uniform Standard deviation**: 𝜎=√((𝑏−𝑎)^2 / 12)**Uniform pdf**: 𝑓(𝑥)=1 / 𝑏−𝑎 for*a ≤ x ≤ b***Uniform cdf**:*P*(*X*≤*x*) = 𝑥−𝑎 / 𝑏−𝑎

*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***Probability density function:**𝑓(𝑥)=(1 / b−a) for 𝑎 ≤ 𝑋 ≤ 𝑏**Area to the Left of****:***x**P*(*X*<*x*) = (*x*–*a*)(1 / 𝑏−𝑎)**Area to the Right of****:***x**P*(*X*>*x*) = (*b*–*x*)(1 / 𝑏−𝑎)**Area Between***c***and****:***d**P*(*c*<*x*<*d*) = (base)(height) = (*d*–*c*)(1 / 𝑏−𝑎)

**Memoryless property**: the independence of events or, more specifically, the independence of event-to-event times or*P*(*X*>*r*+*t*|*X*>*r*) =*P*(*X*>*t*) for all*r*≥ 0 and*t*≥ 0**Exponential Distribution**:*X*~*Exp*(*m*) where*m*= the decay parameter**decay parameter**:*m*= 1 / μ and we write*X*∼*Exp*(*m*) where*x*≥ 0 and*m*> 0**exponential pdf**:*f*(*x*) =*me^*(–*mx*) where*x*≥ 0 and*m*> 0**exponential cdf**:*P*(*X*≤*x*) = 1 –*e^*(–*mx*)**exponential mean**:*µ*= 1/𝑚**exponential standard deviation**:*σ*=*µ***exponential percentile***k*:*k*= 𝑙𝑛(1−𝐴𝑟𝑒𝑎𝑇𝑜𝑇ℎ𝑒𝐿𝑒𝑓𝑡𝑂𝑓𝑘) / (−𝑚)Additionally

*P*(*X*>*x*) =*e^*(–*mx*)*P*(*a*<*X*<*b*) =*e^*(–*ma*) –*e^*(–*mb*)

**Poisson probability**: 𝑃(𝑋=𝑘)=𝜆^𝑘 𝑒^−𝑘 / 𝑘! P(X=k) with mean*λ***!** =*k**k**(*k*-1)*(*k*-2)*(*k*-3)*…3*2*1