Continuous probability distributions
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1 Continuous probability distributions Many continuous probability distributions, including: Uniform Normal Gamma Eponntial Chi-Squard Lognormal Wibull EGR 5 Ch. 6 Uniform distribution Simplst charactrizd by th intrval ndpoints, A and B. f ( ; A, B B A A B = 0 lswhr Man and varianc: A B and ( B A EGR 5 Ch. 6
2 Eampl A circuit board failur causs a shutdown of a computing systm until a nw board is dlivrd. Th dlivry tim X is uniformly distributd btwn and 5 days. What is th probability that it will tak or mor days for th circuit board to b dlivrd? 3 EGR 5 Ch. 6 Normal distribution Th bll-shapd curv Also calld th Gaussian distribution Th most widly usd distribution in statistical analysis forms th basis for most of th paramtric tsts w ll prform latr in this cours. dscribs or approimats most phnomna in natur, industry, or rsarch Random variabls (X following this distribution ar calld normal random variabls. th paramtrs of th normal distribution ar μ and σ (somtims μ and σ. 4 EGR 5 Ch. 6
3 P( Normal distribution Th dnsity function of th normal random variabl X, with man μ and varianc σ, is n( ;, ( all. Normal Distribution (μ = 5, σ = EGR 5 Ch. 6 Standard Normal RV Not: th probability of X taking on any valu btwn and is givn by: P( X n( ;, d ( d To as calculations, w dfin a normal random variabl Z X whr Z is normally distributd with μ = 0 and σ = 6 EGR 5 Ch. 6 3
4 Standard Normal Distribution Tabl A.3: Aras Undr th Normal Curv Standard Normal Distribution Z EGR 5 Ch. 6 Eampls P(Z = P(Z - = P(-0.45 Z 0.36 = EGR 55 Ch. 6 4
5 Your turn Us Tabl A.3 to dtrmin (draw th pictur!. P(Z 0.8 =. P(Z.96 = 3. P(-0.5 Z 0.5 = 4. P(Z -.0 or Z.0 = 9 EGR 5 Ch. 6 Th normal distribution in rvrs Eampl: Givn a normal distribution with μ = 40 and σ = 6, find th valu of X for which 45% of th ara undr th normal curv is to th lft of X. If P(Z < k = 0.45, k = Z = X = EGR 5 Ch. 6 5
6 Normal approimation to th binomial If n is larg and p is not clos to 0 or, or if n is smallr but p is clos to 0.5, thn th binomial distribution can b approimatd by th normal distribution using th transformation: Z ( X 0.5 npq np NOTE: add or subtract 0.5 from X to b sur th valu of intrst is includd (draw a pictur to know which Look at ampl 6.5, pg. 9 EGR 5 Ch. 6 Look at ampl 6.5, pg. 9 p = 0.4 n = 00 μ = σ = if = 30, thn z = and, P(X < 30 = P (Z < = EGR 5 Ch. 6 6
7 Your turn Rfr to th prvious ampl, DRAW THE PICTURE!!. What is th probability that mor than 50 surviv?. What is th probability that actly 45 surviv? 3 EGR 5 Ch. 6 Gamma & ponntial distributions Rcall th Poisson Procss Numbr of occurrncs in a givn intrval or rgion Mmorylss procss Somtims w r intrstd in th tim or ara until a crtain numbr of vnts occur. For ampl An avrag of.7 srvic calls pr minut ar rcivd at a particular maintnanc cntr. Th calls corrspond to a Poisson procss. What is th probability that up to a minut will laps bfor calls arriv? How long bfor th nt call? 4 EGR 5 Ch. 6 7
8 f( Gamma Distribution Th dnsity function of th random variabl X with gamma distribution having paramtrs α (numbr of occurrncs and β (tim or rgion. f ( ( > 0. ( n ( n! Gamma Distribution 0.8 μ = αβ σ = αβ EGR 5 Ch. 6 Eponntial distribution Spcial cas of th gamma distribution with α =. f ( > 0. Dscribs th tim until or tim btwn Poisson vnts. μ = β σ = β EGR 5 Ch. 6 8
9 Eampl An avrag of.7 srvic calls pr minut ar rcivd at a particular maintnanc cntr. Th calls corrspond to a Poisson procss. What is th probability that up to a minut will laps bfor calls arriv? β = α = P(X = 7 EGR 5 Ch. 6 Eampl (cont. What is th pctd tim bfor th nt call arrivs? β = α = μ = 8 EGR 5 Ch. 6 9
10 Your turn Look at problm 6.40, pag EGR 5 Ch. 6 Chi-squard distribution Spcial cas of th gamma distribution with α = ν/ and β =. f ( / ( / > 0. whr ν is a positiv intgr. singl paramtr,ν is calld th dgrs of frdom. μ = ν σ = ν 0 EGR 5 Ch. 6 0
11 Lognormal distribution Whn th random variabl Y = ln(x is normally distributd with man μ and standard dviation σ, thn X has a lognormal distribution with th dnsity function, f ( ;, [ln( ], 0 / ( EGR 5 Ch. 6 Eampl Look at problm 6.7, pg. 07 Sinc ln(x has normal distribution with μ = 5 and σ =, th probability that X > 50,000 is, P(X > 50,000 = EGR 5 Ch. 6
12 Wibull distribution Usd for many of th sam applications as th gamma and ponntial distributions, but dos not rquir mmorylss proprty of th ponntial f ( ;,, 0 F( 3 EGR 5 Ch. 6 Eampl Dsignrs of wind turbins for powr gnration ar intrstd in accuratly dscribing variations in wind spd, which in a crtain location can b dscribd using th Wibull distribution with α = 0.0 and β =. A dsignr is intrstd in dtrmining th probability that th wind spd in that location is btwn 3 and 7 mph. P(3 < X < 7 = 4 EGR 5 Ch. 6
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