Textbook: D.C. Montgomery and G.C. Runger, Applied Statistics and Probability for Engineers, John Wiley & Sons, New York, Sections

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1 Textbook: DC Motgomery ad GC Ruger, Applied Statistics ad Probability for Egieers, Joh Wiley & Sos, New York, 2003 Sectios Defiitio of "radom variable" (a) Write the defiitio of a radom variable < A radom variable is a fuctio that assigs a > < real umber to every outcome i the sample space > (b) Give a example (with a cotext) of a experimet, sample space, ad two correspodig radom variables Experimet: Flip a coi Sample space: { heads, tails } Let X 1 0 if tails ad X 1 1 if heads Let X 2 $ 5 if tails ad X 1 $ 5 if heads Here X 1 correspods to a idicator variable used to cout the umber of heads Here X 2 correspods to a bet that I place with a fried, where I pay $5 if heads ad I receive $5 if tails 2 (Motgomery ad Ruger, problem from Sectio 3 1) A fuctio yields a value for each argumet For example, i algebra the usual otatio y f (x ) meas that the the argumet X returs the value y The set of possible values is the rage ad the set of argumets is the domai What is the rage of these radom variables? (a) N "the umber of ocoformig solder coectios o a prited circuit board with 1000 coectios" {0, 1,, 1000} (b) W "the weight displayed o a electroic scale that shows at most five digits, rouded to the earest poud" {0, 1,, 99,999} (c) T "the temperature (i degrees F) arisig withi a chemical experimet" the iterval [ 459, ), or{x 459 x } 1 of 2 Schmeiser

2 3 (Motgomery ad Ruger, Problem 2 100) Decide whether a discrete or cotiuous radom variable is the best model for each of the followig situatios (a) T "the time util a projectile returs to earth" < cotiuous > (b) N "the umber of cracks exceedig oe-half ich i 10 miles of a iterstate highway" < discrete > (c) X "time spet waitig i lie before the bak teller begis your service" < mixed, discrete at zero ad cotiuous for positive values > Commet: For each of these radom variables, do you kow the uderlyig experimet? A appropriate sample space? 4 Idicator radom variables We previously studied evet probabilities Evets either occur or do ot occur; a radom variable always returs a real umber A idicator radom variable is a way to write a evet as a radom variable Let X 1 if A occurs ad X 0 if A occurs Let p P(A ) (a) Write the pmf o i terms of p 1 p if x 0 (x ) p if x 1 0 elsewhere (b) For p 03, sketch the pmf o (Label ad scale both axes) < Horizotal ad vertical axes > < Label horizotal with x ad mark "0" ad "1" > < Label the vertical with (x ) ad scale with "0" ad "1" > < Two vertical lies above "0" ad "1", with heights > < "1 p 07" ad "p 03", respectively > 2 of 2 Schmeiser

3 (c) For p 03, sketch the cdf o (Label ad scale both axes) < Horizotal ad vertical axes > < Label horizotal with x ad mark "1", "2", ad "3" > < Label the vertical with F X (x ) ad scale with "0" ad "1" > < Horizotal lie from to "1" at height "0" > < Horizotal lie from "1" to "2" at height "1 / 3" > < Horizotal lie from "2" to "3" at height "2 / 3" > < Horizotal lie from "3" to at height "1" > (d) Write the cdf o 0 if x<1 1 / 3 if 1 x<2 F X (x ) 2 / 3 if 2 x<3 1 if 3 x (e) Show that E(X ) p E(X ) 0 (1 p ) + 1 p defiitio of E(X ) p simplify (f) Show that V(X ) p (1 p ) V(X ) E(X 2 2 ) µ X previous result [0 2 (1 p ) p ] µ X defiitio of E(X 2 ) [0 2 (1 p ) p ] p 2 Part (e) p p 2 simplify p (1 p ) simplify (g) What value of p maximizes V(X )? < Take the first derivative of V(X ) with respect to p ad set to zero > < I particular, d V(X ) /dp d (p (1 p )) /dp 1 2p 0, < which implies that p 1 / 2 > < This is a maximum, either by sketchig the fuctio > < or oticig that the secod derivative is egative > Commet: If this problem seems too abstract to iteralize, the substitute a specific cotext For example, let the experimet be to flip a coi ad let A be the set of all outcomes that result i head facig up The for a "fair" coi flip, p 05 3 of 2 Schmeiser

4 5 (Motgomery ad Ruger, Problem 3 34) Errors i a experimetal trasmissio chael are foud whe the trasmissio is checked by a certifier that detects missig pulses Let X "umber of errors foud i a eight-bit byte" Suppose that the distributio o has the followig cdf F (x ) 00 if <x <1 07 if 1 x<4 09 if 4 x<7 10 if 7 x< (a) Sketch the cdf (Label ad scale both axes) < Horizotal ad vertical axes > < Label horizotal with x ad mark "1", "4", ad "7" > < Label the vertical with F X (x ) ad scale with "0" ad "1" > < Horizotal lie from to "1" at height "0" > < Horizotal lie from "1" to "4" at height "07" > < Horizotal lie from "4" to "7" at height "09" > < Horizotal lie from "7" to at height "1" > (b) Write the pmf 07 if x 1 02 if x 4 (x ) 01 if x 7 0 elsewhere (c) Sketch the pmf (Label ad scale both axes) < Horizotal ad vertical axes > < Label horizotal with x ad mark "1", "4", ad "7" > < Label the vertical with (x ) ad scale with "0" ad "1" > < Three vertical lies above "1", "4", ad "7", with respective > < heights of 07, 02, ad 01 > (d) What is the umerical value of P(X 5)? P(X 5) P(X 1) + P(X 4) or P(X 5) F X (5) 09 (a value that is give) 4 of 2 Schmeiser

5 (e) What is the umerical value of F X (5)? < See Part (d) > (f) What is the umerical value of (10 6 )? (10 6 ) 0 (from Part (b)) (g) Determie the value of E(X ) E(X ) Σall x x (x ) (h) Determie the value of V(X ) V(X ) E(X 2 ) µ X Σall x x 2 (x ) (24) 2 [ ] (i) Determie the value of σ X σ X V(X ) Result: I has a discrete uiform distributio with lower boud a ad upper boud b, the E(X ) (a + b ) / 2 Supply reasos for the followig proof Proof Suppose that the rage o is a, a +δ, a + 2δ,,b Let deote the umber of values i the rage The b a + ( 1)δ E(X ) Σallx x i (x ) < defiitio of E(X ) > Σi 1 Σi 1 (1 /) Σi 1 (a + (i 1)δ) (a + (i 1)δ) < substitute specific x values > (a + (i 1)δ)(1/) < substitute give pmf > (1 /)[a +δσi 1 (a + (i 1)δ) < factor the costat outside > (i 1)] < algebra > (1 /)[a +δ ( 1) / 2] < cool result try it > a +δ( 1) / 2 < simplify > a + (b a ) / 2 < defiitio of b > (a + b ) / 2 < simplify > 5 of 2 Schmeiser

Closed book and notes. No calculators. 60 minutes, but essentially unlimited time.

Closed book and notes. No calculators. 60 minutes, but essentially unlimited time. IE 230 Seat # Closed book ad otes. No calculators. 60 miutes, but essetially ulimited time. Cover page, four pages of exam, ad Pages 8 ad 12 of the Cocise Notes. This test covers through Sectio 4.7 of