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Continuous-type random variables
1. Normal (Gaussian): X is said to be normal or Gaussian
r.v, if
This is a bell shaped ...
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Grades of a Class
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Uniform Distribution
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Exponential Distribution
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Triangular Distribution
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Laplace Distribution
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Erlang Distribution
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Gamma Distribution
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Chi Square Distribution
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Discrete-type random variables
1. Bernoulli: X takes the values (0,1), and
2. Binomial: ∼ if (Fig. 3.17)
3. Poisson: ∼ ...
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Multinomial Distribution
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Geometric Distribution
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4. Hypergeometric:
5. Geometric: ∼ if
6. Negative Binomial: ~ if
7. Discrete-Uniform:
We conclude this lecture with a g...
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Probability And Random Variable Lecture6

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This ppt is prepared for engineering class. Random variable and stochastic prosses and probability.

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Probability And Random Variable Lecture6

  1. 1. 1 Continuous-type random variables 1. Normal (Gaussian): X is said to be normal or Gaussian r.v, if This is a bell shaped curve, symmetric around the parameter and its distribution function is given by where is often tabulated. Since depends on two parameters and the notation ∼ will be used to represent (3-29). . 2 1 )( 22 2/)( 2 σµ πσ −− = x X exf (3-29) ,µ , 2 1 )( 22 2/)( 2∫∞− −−       − == x y X x GdyexF σ µ πσ σµ (3-30) dyexG y x 2/2 2 1 )( − ∞−∫= π ),( 2 σµNX )(xfX x µ Fig. 3.7 ∆ )(xfX µ ,2 σ PILLAI
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  4. 4. 4
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  7. 7. 7
  8. 8. 8
  9. 9. 9 Grades of a Class
  10. 10. 10 Uniform Distribution
  11. 11. 11
  12. 12. 12
  13. 13. 13 Exponential Distribution
  14. 14. 14
  15. 15. 15 Triangular Distribution
  16. 16. 16 Laplace Distribution
  17. 17. 17 Erlang Distribution
  18. 18. 18 Gamma Distribution
  19. 19. 19
  20. 20. 20 Chi Square Distribution
  21. 21. 21
  22. 22. 22 Discrete-type random variables 1. Bernoulli: X takes the values (0,1), and 2. Binomial: ∼ if (Fig. 3.17) 3. Poisson: ∼ if (Fig. 3.18) .)1(,)0( pXPqXP ==== (3-43) ),,( pnBX .,,2,1,0,)( nkqp k n kXP knk =      == − (3-44) ,)(λPX .,,2,1,0, ! )( ∞=== − k k ekXP k λλ (3-45) k )( kXP = Fig. 3.17 12 n )( kXP = Fig. 3.18 PILLAI
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  24. 24. 24
  25. 25. 25 Multinomial Distribution
  26. 26. 26
  27. 27. 27 Geometric Distribution
  28. 28. 28 4. Hypergeometric: 5. Geometric: ∼ if 6. Negative Binomial: ~ if 7. Discrete-Uniform: We conclude this lecture with a general distribution due PILLAI (3-49) (3-48) (3-47) .,,2,1, 1 )( Nk N kXP === ),,( prNBX 1 ( ) , , 1, . 1 r k r k P X k p q k r r r − −  = = = + −   .1,,,2,1,0,)( pqkpqkXP k −=∞===  )( pgX , max(0, ) min( , )( ) m N m k n k N n m n N k m nP X k                       − − + − ≤ ≤= = (3-46)
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