[ ] ( ) ( ) [ ] ( ) 1 [ ] [ ] Sums of Random Variables Y = a 1 X 1 + a 2 X 2 + +a n X n The expected value of Y is:

Transcription

1 PROBABILITY FUNCTIONS A radom variable X has a probabilit associated with each of its possible values. The probabilit is termed a discrete probabilit if X ca assume ol discrete values, or X = x, x, x 3,, x The discrete probabilit of a sigle evet, X = x i, occurrig is defied as P(x i ) while the probabilit mass fuctio of the radom variable X is defied b f ( xk ) = P( X = xk ), k =,,..., Probabilit Desit Fuctio If X is cotiuous, the probabilit desit fuctio, f, is defied such that P ( a X b) = f ( x) dx b a Cumulative Distributio Fuctios The cumulative distributio fuctio, F, of a discrete radom variable X that has a probabilit distributio described b P(x i ) is defied as m m k= k m F( x ) = P( x ) = P( X x ), m=,,..., If X is cotiuous, the cumulative distributio fuctio, F, is defied b x ( ) = ( ) F x f t dt which implies that F(a) is the probabilit that X a. Expected Values Let X be a discrete radom variable havig a probabilit mass fuctio f ( xk ), k =,,..., The expected value of X is defied as [ ] ( ) µ= E X = x f x k= The variace of X is defied as k k= k [ ] ( ) ( ) = V X = x µ f x Let X be a cotiuous radom variable havig a desit fuctio f (X) ad let Y = g(x) be some geeral fuctio. The expected value of Y is: [ ] [ ] E Y = E g( X) = g( x) f( x) dx k k ENGINEERING PROBABILITY AND STATISTICS (cotiued) The mea or expected value of the radom variable X is ow defied as [ ] ( ) µ= E X = xf x dx while the variace is give b = V [ X ] = E [( X µ ) ] = The stadard deviatio is give b = V[ X] ( x µ ) The coefficiet of variatio is defied as /µ. Sums of Radom Variables Y = a X + a X + +a X The expected value of Y is: f ( x ) dx ( Y ) = a E( X ) + a E( X ) +... a E( X ) µ = E + If the radom variables are statisticall idepedet, the the variace of Y is: = V = a ( Y ) = a V ( X ) + a V ( X ) a V ( X ) + a a Also, the stadard deviatio of Y is: = Biomial Distributio P(x) is the probabilit that x successes will occur i trials. If p = probabilit of success ad q = probabilit of failure = p, the x x! x x P ( x) = Cxpq (, ) = pq, x! ( x)! where x = 0,,,,, C(, x) = the umber of combiatios, ad, p = parameters. Normal Distributio (Gaussia Distributio) This is a uimodal distributio, the mode beig x = µ, with two poits of iflectio (each located at a distace to either side of the mode). The averages of observatios ted to become ormall distributed as icreases. The variate x is said to be ormall distributed if its desit fuctio f (x) is give b a expressio of the form x µ f ( x) = e, where π µ = the populatio mea, = the stadard deviatio of the populatio, ad x 6

2 Whe µ = 0 ad = =, the distributio is called a stadardized or uit ormal distributio. The It is oted that f x ( x) = e, where x. π x µ Z = follows a stadardized ormal distributio fuctio. A uit ormal distributio table is icluded at the ed of this sectio. I the table, the followig otatios are utilized: F(x) = the area uder the curve from to x, R(x) = the area uder the curve from x to, ad W(x) = the area uder the curve betwee x ad x. The Cetral Limit Theorem Let X, X,..., X be a sequece of idepedet ad ideticall distributed radom variables each havig mea µ ad variace. The for large, the Cetral Limit Theorem asserts that the sum Y = X+ X +... X is approximatel ormal. t-distributio The variate t is defied as the quotiet of two idepedet variates x ad r where x is uit ormal ad r is the root mea square of other idepedet uit ormal variates; that is, t = x/r. The followig is the t-distributio with degrees of freedom: Γ () [( + ) ] f t = ( ) ( ) ( + Γ π + t ) where t. A table at the ed of this sectio gives the values of t α, for values of α ad. Note that i view of the smmetr of the t-distributio, t α, = t α,. The fuctio for α follows: α = tα, χ - Distributio f () t dt If Z, Z,..., Z are idepedet uit ormal radom variables, the Z Z... Z χ = is said to have a chi-square distributio with degrees of freedom. The desit fuctio is show as follows: f χ e χ ( ) χ =, χ > 0 Γ ( ) A table at the ed of this sectio gives values of χ α, for selected values of α ad. ENGINEERING PROBABILITY AND STATISTICS (cotiued) Gamma Fuctio Γ t ( ) t e dt, > 0 = µ =µ o ad the stadard deviatio = LINEAR REGRESSION Least Squares = â + bˆx, where - itercept : a ˆ = bx ˆ, ad slope : b ˆ = S /S, ( ) S = x / x, x i i i i i= i= i= S x ( /) x, = xx i i i= i= = sample size, = ( /), ad i i= x = ( /) x. i i= x xx Stadard Error of Estimate S S S xxs S x e = = S xx ( ) i = i ( ) i= i= Cofidece Iterval for a MSE, where x â t, MSE S ± α + xx Cofidece Iterval for b bˆ ± t α, MSE S Sample Correlatio Coefficiet S x r = S S xx xx 7

3 HYPOTHESIS TESTING Cosider a ukow parameter θ of a statistical distributio. Let the ull hpothesis be H 0: θ=θ0 ad let the alterative hpothesis be H : θ=θ ENGINEERING PROBABILITY AND STATISTICS (cotiued) Rejectig H 0 whe it is true is kow as a tpe I error, while acceptig H whe it is wrog is kow as a tpe II error. Furthermore, the probabilities of tpe I ad tpe II errors are usuall represeted b the smbols α ad β, respectivel: α=probabilit (tpe I error) β=probabilit (tpe II error) The probabilit of a tpe I error is kow as the level of sigificace of the test. Assume that the values of α ad β are give. The sample size ca be obtaied from the followig relatioships. I (A) ad (B), µ is the value assumed to be the true mea. (A) H : µ=µ ; H : µ µ µ 0 µ µ 0 µ β=φ +Ζα Φ Ζα A approximate result is ( α β) ( µ µ ) Ζ +Ζ 0 (B) H : µ=µ ; H : µ>µ µ 0 µ β=φ +Ζ = ( α β) ( µ µ ) 0 Ζ +Ζ α Refer to the Hpothesis Testig table i the INDUSTRIAL ENGINEERING sectio of this hadbook. 8

4 CONFIDENCE INTERVALS Cofidece Iterval for the Mea µ of a Normal Distributio (a) Stadard deviatio is kow X Zα µ X + Zα (b) Stadard deviatio is ot kow s s X tα µ X + tα where t α correspods to degrees of freedom. ENGINEERING PROBABILITY AND STATISTICS (cotiued) Cofidece Iterval for the Differece Betwee Two Meas µ ad µ (a) Stadard deviatios ad kow α + µ µ + α + X X Z X X Z (b) Stadard deviatios ad are ot kow ( ) S ( ) S ( ) S ( ) S X X tα µ µ X X tα + + where tα correspods to + degrees of freedom. Cofidece Itervals for the Variace of a Normal Distributio ( ) ( ) α, x α, s s x Sample Size X µ z = Value of Z α Cofidece Iterval Z α 80%.86 90% % % % %

5 ENGINEERING PROBABILITY AND STATISTICS (cotiued) 0 UNIT NORMAL DISTRIBUTION x f(x) F(x) R(x) R(x) W(x) Fractiles

6 STUDENT'S t-distribution ENGINEERING PROBABILITY AND STATISTICS (cotiued) VALUES OF t α, α = 0.0 α = 0.05 α = 0.05 α = 0.0 α = α

7 CRITICAL VALUES OF THE F DISTRIBUTION For a particular combiatio of umerator ad deomiator degrees of freedom, etr represets the critical values of F correspodig to a specified upper tail area (α). Numerator df Deomiator df ENGINEERING PROBABILITY AND STATISTICS (cotiued)

8 CRITICAL VALUES OF X DISTRIBUTION f( X ) α 0 X α, X 3 Degrees of Freedom X.995 X.990 X.975 X.950 X.900 X.00 X.050 X.05 X.00 X Source: From Thompso, C. M. Tables of the Percetage Poits of the X -Distributio, Biometrika, 94, 3, Reproduced b permissio of the Biometrika Trustees ENGINEERING PROBABILITY AND STATISTICS (cotiued)