3. Several Random Variables
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1 . Several Random Variables. Two Random Variables. Conditional Probabilit--Revisited. Statistical Independence.4 Correlation between Random Variables. Densit unction o the Sum o Two Random Variables. Probabilit Densit unction o a unction o Two Random Variables.7 The Characteristic unction Concepts Two Dimensional Random Variables Probabilit in Two Dimensions Conditional Probabilit--Revisited Statistical Independence Two Dimensional Statistics Correlation between Random Variables Densit unction o the Linear Combination o Two Random Variables Multi-input lectrical Circuits Simulating Convolution Integrals Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring o 8 C 8
2 Joint Probabilit Distribution unction (PD) Probabilit Distribution unction:the probabilit o the event that the observed random variable is less than or equal to the allowed value and that the observed random variable is less than or equal to the allowed value. Pr The deined unction can be discrete or continuous along the - and -ais. Constraints on the probabilit distribution unction are:. or and.. 4. is non-decreasing as either or increases. and Analogies: a -dimensional probabilit moving rom scalars to vectors ( or more elements) Calc as compared to Calc & Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring o 8 C 8
3 Joint Probabilit Densit unction (pd) The derivative o the probabilit distribution unction is the densit unction Properties o the pd include. or and. d d Note: the volume o the -D densit unction is one.. u v du dv 4. d and d. Pr d d Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring o 8 C 8
4 pected Values g g d d All epected values ma be computed using the Joint pd Correlation The deinition o a new epected value Correlation d d A value describing the relationship how correlated two random variables are to each other. Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring 4 o 8 C 8
5 Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring o 8 C 8 Uniorm Densit ample The uniorm densit unction in two dimensions can be deined as: else and or Determine the densit in d d or Similarl or Correlation d d d d d d 4
6 Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring o 8 C 8 ercise -. Part a and b would appear to have the same result. It would seem that the random errors should be independent so parts the nd and rd solution values make sense. Where the irst solution value came rom I do not know. ercise -. else and or A ep Determine A ep d d A d d ep ep ep ep d A d d A ep ep A A d A A Determine the Distribution unction d d ep d d ep ep d ep ep ep ep ep ep Then 4 ep ep 4 inall determine the correlation
7 Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring 7 o 8 C 8 d d ep d d Given ep ep a a a a 9 ep 4 ep
8 Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring 8 o 8 C 8 Conditional Probabilit (Again with multiple r.v.) Using the Probabilit Distribution unction (PD) deine M M Pr Pr Another wa. Leading to and These are dierent rom the probabilit o a continuous distribution taking on a single value in and or An engineering derivation ollows: lim lim du u [Note: qu(-9) is in error the integral is to not ininit] Then
9 The corresponding conditional densit unction is and similarl it can be shown that [Note: qu (-) is in error the denominator is a variable in.] rom these equations it can be seen that The joint densit total probabilit concepts can deine the and marginal densities. d and d Then rom the conditional densit relationship with the joint densit we can replace the joint densit unctions in the total probabilit equations to deine the pd densities o and based on the conditional densities as or d d d d Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring 9 o 8 C 8
10 To derive the multiple variable Baes Theorem use resulting in or Note: the joint probabilit densit unction completel speciies: both marginal densit unctions and both conditional densit unctions. Warning: The eample on 7 and 8 does not make sense without inormation rom p. -7. In particular q. -7 implies that ou know how to orm () when this concept has not been introduced. It should be revisited ater discussing the pd o the sum o two random variables. Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring o 8 C 8
11 Statistical Independence Wh have alread seen this in the two eamples presented. Joint Uniorm Densities or and Where we proved and or or Thereore ercise -. or and Where it turns out that A ep or and ep ep or and As and d ep ep ep ep d ep ep ep d ep Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring o 8 C 8
12 Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring o 8 C 8 Deinition o correlation or and independent d d d d As another consequence the conditional densit is simpliied as and similarl [Note: p. irst equation is in error the denominator is the densit unction.] In general i independence can be establish or even assumed the computations to be perormed become much easier!
13 Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring o 8 C 8 ample p. revisited or independence and correlation. and or Where the marginal densities are d and d Note that Thereore the variables are not independent! rom computations: 7. and 9. 7 Then the correlation value is d d d d d 4 4 d 4 And again which would be the case or independents random variables.
14 ample -. k ep or and There is no a unless the problem was supposed to be stated as k ep a or and but then k and a are not necessaril computed separatel as and.! Overall the correct pd is Correlation ep or and d d ep d d a a ep epa d a ep ep ep ep ep d d ep ep ep ep Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring 4 o 8 C 8
15 ercise -. Assume and independent ind Pr. ep or. ep or? That is the product o the random variables is positive. Pr Pr Pr Pr Pr Pr ind the distribution or and based on the ranges deined or the absolute value or or and or. ep d or or. ep d ep.. ep.. ep ep.. d.. ep Pr. ep Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring o 8 C 8
16 Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring o 8 C 8 Correlation and Covariance between Random Variables The deinition o correlation was given as d d But most o the time we are not interested in products o mean values (observed when and are independent) but what results when the are removed prior to the computation. Developing values where the random variable means have been etracted is deined as computing the covariance d d This gives rise to another actor when the random variable variances are used to normalize the actors or covariance computation as: d d This equation deines the correlation coeicient or normalized covariance the modiied random variables are called the standardized variables and have zero mean and a unit variance. An alternate epression or the correlation coeicient is derived b perorming the multiplication d d d d d d d d d d
17 Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring 7 o 8 C 8 d d d d d d An alternate method to skip the integrals The epected value is a linear operator constants remain constants and sums are sums Simpliications or random variables that are inherentl zero mean with a unit variance or either or a zero mean variable and or independent random variables
18 Notes and igures are based on or taken rom materials in the course tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bazuin Spring 8 o 8 C 8 Standardized variables and have zero mean and a unit variance. This is similar to using the normal densit/distribution or a Gaussian. The standardized or normalized R.V. must have a zero mean and unit variance (normalized). and Note that now and and Remember: this is generalized or all R.V not just Gaussian/Normal R.V. There are also computations based on the sum and dierence o these random variables that can be computed. and the variance is the same value Var
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