Recall Discrete Distribution. 5.2 Continuous Random Variable. A probability histogram. Density Function 3/27/2012

Transcription

1 3/7/ Recall Discrete Distribution 5. Continuous Random Variable For a discrete distribution, for eample Binomial distribution with n=5, and p=.4, the probabilit distribution is f() A probabilit histogram How to describe the distribution of a continuous random variable? For continuous random variable, we also represent probabilities b areas not b areas of rectangles, but b areas under continuous curves. P()... For continuous random variables, the place of histograms will be taken b continuous curves. Imagine a histogram with narrower and narrower classes. Then we can get a curve b joining the top of the rectangles. This continuous curve is called a probabilit densit (or probabilit distribution). Continuous distributions: Densit Function For an, P(X=)=. (For a continuous distribution, the area under a point is.) Can t use P(X=) to describe the probabilit distribution of X Instead, consider P(a X b) Densit Function A probabilit densit function for a continuous random variable X is a nonnegative function f() with + f( d ) = And such that for all a b, one is willing to assign P[a X b] according to b Pa [ X b] = f( d ) a

2 3/7/ Densit function P( X 4)= P( X<4)= P(<X<4) A curve f(): f() The area under the curve is P(a X b) is the area between a and b Cumulative Probabilit function For X continuous with probabilit densit f() F( ) = PX [ ] = f() tdt f( ) = e, > t F( ) = e dt = e We can get the densit function f() from F() b differentiation d F ( ) = f ( ) d conversel d f( ) = ( e ) = e d

3 3/7/ The normal distribution A normal curve: Bell shaped Densit is given b ( µ ) f ( ) = ep σ π σ μand σ are two parameters: mean and variance of a normal population (σ is the standard deviation) The normal Bell shaped curve: μ=, σ = Normal curves: (μ=, σ =) and (μ=5, σ =) f f

4 3/7/ Normal curves: (μ=, σ =) and (μ=, σ =) Normal curves: (μ=, σ =) and (μ=, σ =.5) f The standard normal curve: μ=, and σ = Table B.3 gives probabilities for standard normal (Numerical Integration No formula for From Table B.3 = Eample: If, then 4

5 fz 3/7/ Eample (p33 #7) In a grinding operation, there is an upper specification of 3.5 in. on a dimension of a certain part after grinding. Suppose that the standard deviation of this normall distributed dimension for parts of this tpe ground to an particular mean dimension μ is σ=. in. Suppose further that ou desire to have no more than 3% of the parts fail to meet specifications. What is the maimum μ (minimum machining cost) that can be used if this 3% requirement is to be met? PX> ( 3.5) =.3 So 3.5 is.88 σ above the mean *.= PZ> ( ) =.3 PZ> (.88) = z Eponential distribution The eponential distribution is a continuous probabilit distribution with f e / ( ) =, > Eponential distributions are often used to describe waiting times until occurrence of events. f( ) = e, > Densit curves dep(,.5) f e / ( ) =, > alpha= alpha= 5

6 3/7/ Mean and Variance Mean of an eponential distribution is Cumulative Probabilit Function F( ) = e / / µ = EX ( ) = e d = Variance of the eponential distribution is σ = Var( X) = ( ) e d = / pep(, ) pep(,.5) alpha= alpha= Relationship between Eponential Distribution and Poisson PX ( < ) = F() = e =.8647 When alpha= Ships arrive / hour. The number of ships arriving in hour is a Poisson random variable with λ= P(X<) on densit curve f() P(X<) on CDF F() pep(, ) Let s define a new random variable X to be the waiting time until the first ship. P(X )=P( ships b time ) F()=P(X ) =-P( ships b time ) alpha= An eponential distribution with mean has f( ) = e / As above with Memorless propert: If we have alread waited H hours and haven t seen a ship, our epected waiting time is. Our epected waiting time is like starting all over again. The probabilit of a ship showing up in the net 5 minutes is the same as when we started. 6

7 3/7/ Force of Mortalit Function The force of Mortalit Function is (p.76): f() t f() t ht () = = PT ( > t) Ft () H(t)dt is the probabilit of ding in time t to t+dt if we are still living in t. For eponential distribution / e f () t f () t ht () = = = = / PT ( > t) Ft () ( e ) So the eponential distribution has a constant force-of-mortalit. If the lifetime of an engineering component is described using a constant force of mortalit, there is no (mathematical) reason to replace such a component before it fails. The distribution of its remaining life from an point in time is the same as the distribution of the time till failure of a new component of the same tpe. Section 5..3 WeibullDistribution The geometric distribution is also memorless. X = time to success. The epected tosses to net head doesn t depend on how long we have been tossing without getting a head. Lifetimes of glasses in a restaurant might be eponential. Motor lifetimes or people s lifetimes are not. Sometimes lifetimes can be modeled with a lognormal distribution where Ver commonl lifetimes of motors, etc. are modeled with Weibull distributions. A Weibull distribution is a generalization of an eponential distribution and provides more fleibilit in terms of distributional shape. For Weibulldistribution F( ) = e β ( / ) d β f( ) = F( ) = e β d µ = E( X) = Γ ( + ) β β β ( / ) Gamma function For the force-of-mortalit is a decreasing function, for eample, a product break in a period. Constant force-of-mortalit. Eponential distribution Increasing force-of-mortalit, for eample, a product that wears out. 7

8 3/7/ Eercise For component with increasing force-ofmortalit (IFM) distribution, such components are retired from service once the reach a particular age, even if the have not failed. The lifetime of a certain tpe of batter has an eponential distribution with average lifetime hours. 5 batteries are installed at the same time and suppose that the operations of the batteries are independent. Find the probabilit that onl batteries are still working after 5 hours. 8