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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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9. 9. 9 Grades of a Class
10. 10. 10 Uniform Distribution
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13. 13. 13 Exponential Distribution
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15. 15. 15 Triangular Distribution
16. 16. 16 Laplace Distribution
17. 17. 17 Erlang Distribution
18. 18. 18 Gamma Distribution
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20. 20. 20 Chi Square Distribution
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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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25. 25. 25 Multinomial Distribution
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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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