Chapter 6 Continuous Random Variables and Distributions


 Harry Charles Stafford
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1 Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete rndom vriles. In ddition mesures o time distnce nd temperture it into the sme ctegory nd they re ll represented y continuous rndom vriles. Proility sttements or continuous vriles cn e eplined y continuous distriution unctions. Cumultive Distriution unction Let X e discrete or continuous rndom vrile nd the cumultive distriution unction F X gives the proility tht X does not eceed the vlue o nd given y F X P X Rememer tht or ech discrete rndom vrile X we hd proility distriution unction whose histogrm representtion consists o percentges. points mrked on horizontl is shows the vlues tht X cn tke nd heights o rectngles gives the corresponding proilities. I you rrnge intervls round k s in such wy tht ll they re equl nd hs length then we cn mesure proilities y the res o rectngles. e.g. P X is the re o the shded region elow.
2 nd P X is the re o the shded region elow. Similrly to ind the proility o continuous rndom vriles we need unction clled Proility Density Function which should stisy: or ll vlues o. The totl re under the proility density unction is equl to. The cumultive distriution unction F is the re under the proility density unction up to F m d Where m is the minimum vlue o the rndom vrile. P X F F is the re o the region ounded y the y the grph o nd the horizontl is rom to. d Remrk: I X is continuous rndom vrile then P X or ll i.e. the proility o single vlue is lwys equl to zero. So the ollowing sttement is correct or ll continuous rndom vriles. P X P P X P X The Uniorm Distriution
3 I continuous rndom vrile X hs constnt proility or ll vlues o nd the proility distriution unction is given s ollows X is sid to e Uniormly distriuted over the intervl. In generl X is uniorm rndom vrile on the intervl i its proility density unction is given y And the distriution unction o uniorm rndom vrile on the intervl is given y F Emple : I continuous rndom vrile hs the ollowing proility density unction. / Find X P? Find the proility tht X is greter thn? c Find the proility tht X is greter thn or equsl to? So the cumultive distriution unction is s ollows. + F Now we cn clculte the proilities.
4 P X F F P X > F F P X P X > F F Emple 6.7: The incomes o ll milies in prticulr suur cn e represented y continuous rndom vrile. It is known tht the medin income or ll milies in this suur is $6 nd tht % o ll milies in the suur hve incomes ove $7. For rndomly chosen mily wht is the proility tht its income will e etween $6 nd $7. I the distriution unction or the income is known to e uniorm wht is the proility tht rndom chosen mily hs n income elow $6. We know tht the totl re elow proility density unction or continuous rndom vrile is equl to. Moreover we know tht the medin is the vlue which determines the middle % o the proility density unction. So we cn write tht P X 6 nd P X 7 nd P 6 X 7 P X 6 P X 7 I the distriution unction o the income is Uniorm the shpe o the proility density unction or this continuous rndom vrile X should e s ollows $6 $7 From this chrt we cn sy tht.7 6 so we cn ind the vlue o proility density unction /. Now we cn clculte the proility tht rndom chosen mily hs n income elow $6. P X 6 P X 6 + P 6 X 6 +./ Epecttions or Continuous Rndom Vriles
5 Suppose tht rndom eperiment gives results tht cn e represented y rndom continuous vrile. And i g is ny unction o the rndom vrile X then the epected vlue o X is denoted y E g X nd clculted y s ollows. [ g ] E g d For continuous rndom vriles the men o X is denoted y µ nd µ E X. The vrince is denoted y stndrd devition is. nd E X µ E X µ. And the Emple: Suppose tht X is continuous rndom vrile with the proility density unction. A Find the proility X is more then. B Find the epected vlue o X. C Find the epected vlue o YX+. D Find the vrince nd the stndrd devition o X..d... P X > µ E X d.d. E Y E X + E X E X µ d
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