38.2. The Uniform Distribution. Introduction. Prerequisites. Learning Outcomes

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1 The Uniform Distribution 8. Introduction This Section introduces the simplest type of continuous probbility distribution which fetures continuous rndom vrible X with probbility density function f(x) which ssumes constnt vlue over finite intervl. Prerequisites Before strting this Section you should... Lerning Outcomes On completion you should be ble to... understnd the concepts of probbility be fmilir with the concepts of expecttion nd vrince be fmilir with the concept of continuous probbility distribution explin wht is ment by the term uniform distribution clculte the men nd vrince of uniform distribution 8 HELM (008): Workbook 8: Continuous Probbility Distributions

2 . The uniform distribution The Uniform or Rectngulr distribution hs rndom vrible X restricted to finite intervl [, b] nd hs f(x) constnt over the intervl. An illustrtion is shown in Figure : b f(x) b x Figure The function f(x) is defined by: b, x b Men nd vrince of uniform distribution Using the definitions of expecttion nd vrince leds to the following clcultions. As you might expect, for uniform distribution, the clcultions re not difficult. Using the bsic definition of expecttion we my write: E(X) = = b (b ) = b + xf(x) dx = b Using the formul for the vrince, we my write: V(X) = E(X ) [E(X)] b ( ) = x b +. b dx = ( ) = b b + (b ) = b + b + b + b + 4 (b ) = x [ ] b b dx = x (b ) [ ] b x (b ) ( b + ) HELM (008): Section 8.: The Uniform Distribution 9

3 Key Point The Uniform rndom vrible X whose density function f(x) is defined by b, x b hs expecttion nd vrince given by the formule E(X) = b + nd V(X) = (b ) Exmple The current (in ma) mesured in piece of copper wire is known to follow uniform distribution over the intervl [0, 5]. Write down the formul for the probbility density function f(x) of the rndom vrible X representing the current. Clculte the men nd vrince of the distribution nd find the cumultive distribution function F (x). Solution Over the intervl [0, 5] the probbility density function f(x) is given by the formul = 0.04, 0 x Using the formule developed for the men nd vrince gives E(X) = (5 0) =.5 ma nd V(X) = = 5.08 ma The cumultive distribution function is obtined by integrting the probbility density function s shown below. x f(t) dt Hence, choosing the three distinct regions x < 0, 0 x 5 nd x > 5 in turn gives: 0, x < 0 x 5 0 x 5 x > 5 0 HELM (008): Workbook 8: Continuous Probbility Distributions

4 Tsk The thickness x of protective coting pplied to conductor designed to work in corrosive conditions follows uniform distribution over the intervl [0, 40] microns. Find the men, stndrd devition nd cumultive distribution function of the thickness of the protective coting. Find lso the probbility tht the coting is less thn 5 microns thick. Your solution Answer Over the intervl [0, 40] the probbility density function f(x) is given by the formul { 0.05, 0 x 40 Using the formule developed for the men nd vrince gives E(X) = 0 µm nd σ = V(X) = 0 = 5.77 µm The cumultive distribution function is given by x f(x) dx Hence, choosing pproprite rnges for x, the cumultive distribution function is obtined s: 0, x < 0 x 0 0 x 40 0 x 40 Hence the probbility tht the coting is less thn 5 microns thick is F (x < 5) = = 0.75 HELM (008): Section 8.: The Uniform Distribution

5 Exercises. In the mnufcture of petroleum the distilling temperture (T C) is crucil in determining the qulity of the finl product. T cn be considered s rndom vrible uniformly distributed over C to 00 C. It costs C to produce gllon of petroleum. If the oil distills t tempertures less thn 00 C the product sells for C per gllon. If it distills t temperture greter thn 00 C it sells for C per gllon. Find the expected net profit per gllon.. Pckges hve nominl net weight of kg. However their ctul net weights hve uniform distribution over the intervl 980 g to 00 g. Answers. () Find the probbility tht the net weight of pckge is less thn kg. (b) Find the probbility tht the net weight of pckge is less thn w g, where 980 < w < 00. (c) If the net weights of pckges re independent, find the probbility tht, in smple of five pckges, ll five net weights re less thn wg nd hence find the probbility density function of the weight of the heviest of the pckges. (Hint: ll five pckges weigh less thn w g if nd only if the heviest weighs less tht w g). P(X < 00) = = P(X > 00) = Let F be rndom vrible defining profit. F cn tke two vlues (C C ) or (C C ). x C C C C P(F = x) / / [ ] C C E(F ) = + [C C ] = C C + C () The required probbility is P(W < 000) = = 0 = 0.4 (b) The required probbility is P(W < w) = w = w 980 ( ) 5 w 980 (c) The probbility tht ll five weigh less thn w g is so the pdf of the heviest is ( d w 980 dw ) 5 = 5 ( ) 4 ( ) 4 w 980 w 980 = 0. for 980 < w < 00. HELM (008): Workbook 8: Continuous Probbility Distributions