The number (r) of success in a total of (n) trials is then given by:

Transcription

1 6.3.. Som typical probability distribution functions 6.3..a. Binominal distribution This distribution is rstrictd to random variabls that can only hav two possibl outcoms: succss (or accptabl) or fail (or unaccptabl). Th following conditions hav to b satisfid in ordr to apply a binominal distribution: - Only two possibl outcoms. - Th probability of succss rmains constant throughout th xprimnt. 3- Th xprimnt consists of n indpndnt trials. Th numbr (r) of succss in a total of (n) trials is thn givn by: r) p r p r ( ) n r With r r n!!( n r)! is n combination r. W can gt th xpctd numbr of succsss in n trials for a binomial distribution as: μ np σ np p And standard dviation for th binomial distribution of: ( ) Now, if your objctiv is to gt th probability of an vnt happning lss or qual a crtain numbr of tims ( n ), this is givn as: r ) i 0 r i) i 0 p i i ( p) ni Instrumntation and Masurmnts \ LK\

2 6.3..b. Poisson distribution It is usd to stimat th numbr of random occurrncs of an vnt in a spcifid intrval of tim or spac if th avrag numbr of occurrncs is alrady nown. Hr ar som conditions satisfid by Poisson distribution: - Th probability of occurrnc of any vnt is th sam for two intrvals of th sam lngth. - Th probability of th occurrnc of an vnt is indpndnt of th occurrnc of othr vnts. x) x λ x! λ is th xpctd (or man) numbr of vnts during th intrval of intrst. λ E (x) μ λ and σ λ To dtrmin th probability of a crtain numbr of vnts is lowr than : x ) λ λ i i 0 i! Instrumntation and Masurmnts \ LK\

3 6.3..c. Normal distribution (Gaussian distribution) It is usd whn th valus obtaind ar du uniquly to random factors and occurrncs of both positiv and ngativ dviations ar qually probabl. f ( x) σ π ( xμ ) / σ Whr: x is th random variabl; μ is th population man and σ is th population standard dviation. Figur 6.3. Gaussian (normal) distribution. For a Gaussian distribution, th probability of having a singl valu x btwn two valus x and x is: x x x ) x x f ( x) dx x x σ π ( xμ ) / σ dx Sinc this intgral can not b solvd analytically, it is prformd numrically. For this purpos, a chang of variabl will ma our lif asir. z x μ σ Lading to: f ( z) dnsity function. π z / for σ. This function is calld th standard normal Instrumntation and Masurmnts \ LK\

4 Figur 6.4. Standard normal dnsity function. W can writ now th abov intgral in trms of z: x x x ) z z z ) z z f ( z) dz Th probability that a masurmnt will fall in on of mor standard dviations is summarizd blow: Confidnc Intrval Confidnc Lvl ± σ 68% ± σ 95.44% ± 3 σ 99.74% ± 3.5 σ 99.96% Instrumntation and Masurmnts \ LK\

5 Exampl Automobils visit inspction station an avrag of 40 tims in an ight-hour worday. What is th probability that mor than fiv automobils will visit th inspction station in a on-hour priod? Exampl Th lighting dpartmnt of a city has installd 000 lctric lamps with an avrag lif of 000 h and a standard dviation of 00 h. a) aftr what priod of lightning hours would you xpct 0% of th lamp to fail? b) how many lamps may b xpctd to fail btwn 900 and 300 h? Exampl An lctronic production lin yilds 95% satisfactory parts. Assum that th quality of ach part is indpndnt of th othrs. Four parts ar tstd. Dtrmin th probability that: a) all parts tstd ar satisfactory. b) At last two parts ar satisfactory Paramtr stimation Th main principl bhind statistical mthods is us a small sampl and xtract from it information on th population. Howvr, th sampl is usually small and as a consqunc diffrnt sampls might lad to diffrnt mans and standard dviations for th population. It is, thrfor, ssntial to dtrmin th uncrtainty of th paramtrs xtractd from th sampl. Instrumntation and Masurmnts \ LK\